The proof of [OAI26] regarding Conjecture A follows from the following observation.
Algebraic Topology
Lemma 2.1
Let $X$ be a derived $p$-complete spectrum, that is $X\simeq \holim_n (X/p^n)$, where $X/p^n:=\mathrm{hocofib}(p^n:X\to X)$. Then $\pi_m X$ is $\Z_p$-finitely generated for every $m\in \Z$ if and only if $\pi_m (X/p)$ is finite.
proof:
The exact sequence $X\xrightarrow{p} X\to X/p\xrightarrow{\delta} \Sigma X$ determines an exact sequence for every $m$
\[ 0\to (\pi_m X)/ (p \pi_m X)\to \pi_m X/p\to \pi_{m-1} X[p]\to 0 \]Where $M[p]$ denoted the $p$-torsion in the abelian group $M$.
($\implies$) For every finitely generated $\Z_p$-module $M$ and $N$, $M/pM$ and $N[p]=\mathrm{span}(u_i)$ such that $pu_i=0$ are both finite. Hence, the middle term must be finite.
($\impliedby$) We have exact sequences $X/p^n\xrightarrow{p} X/p^{n+1}\to X/p$. Induction on $n$ shows that if $\pi_mX/p$ is finite, then $\pi_m X/p^n$ must all be finite.
We have Milnor’s $\lim^1$ sequence and derived $p$-completeness of $X$ determines exact sequence \[ 0\to \lim^1\pi_{m-1}(X/p^n) \to \pi_m(X)\to \lim \pi_m(X/p^n)\to 0 \] Since $\pi_{m-1}(X/p^n)$ are finite, and since inverse system of finite abelian groups satisfies Mittag-Leffler condition, $\lim^1$ term vanishes, and $\pi_m(X)\cong \lim \pi_m(X/p^n)$.
Looking at the exact sequence deduced from $X\to X\to X/p$, the left term is a $p$-torsion finite group, the right term is a $p$-torsion finite group. Hence the middle term must be a finite $p$-group (not necessarily a $p$-torsion, as it might not split, but it is ok). Therefore, $\pi_m(X)$ has the $\Z_p$-module structure induced from the inverse limit.
Since $\pi_m(X/p^n)$ are finite, $\pi_m(X)$ is compact with the profinite topology. Moreover, since $\pi_m(X)/p\pi_m(X)$ and $\Z_p$ is a local ring with maximal ideal $p\Z_p$, then by topological Nakayama’s lemma, $\pi_m(X)$ is finitely generated [NSW08, Lemma 5.2.18].
QED.
We have that $K(n)$-local spectrum are derived $p$-complete. Hence, from Lemma 2.1, if we want to show that $\pi_*\LKn\S$ is $\Z_p$-finitely generated, then we just need to show that $(\LKn \S)/p\simeq \LKn (\S/p)$ has finite homotopy group.
Continuous Cohomology
We have \[ 0\to (E_n)_*\S\xrightarrow{(E_n)_*(p)=\times p}(E_n)_*\S\to (E_n)_*\S/p\to 0 \] exact, and therefore, $(E_n)_*\S/p\cong (E_n)_*/p$.
By DHSS, we have \[ E_2^{s,t}=\Hcts^s(\Gn, (E_n)_t/p)\implies \pi_{t-s}\LKn \S/p \]
Since $\Gn=Gal(\Fpn/\Fp)\ltimes P_n$, for every $\Gn$-module $M$. There is the continuous Lyndon-Hochschild-Serre spectral sequence \[ E_2^{p,q}=\Hcts^p(Gal(\Fpn/\Fp), \Hcts^q(P_n, M))\implies \Hcts^{p+q}(\Gn, M) \]
We have $Gal(\Fpn/\Fp)\cong C_{n}$. If a $C_{n}$-module $N$ is finite, then $H^p(C_{n}, N)$ are all finite. Then the god-given vertical and horizontal vanishing line implies that if $\Hcts^q(P_n, M)$ are finite, $\Hcts^{p+q}(\Gn, M)$ are finite.
Lemma 2.2
For spectrum $X$ such that DHSS is strongly convergent, there is a horizontal vanishing line on a finite page of DHSS.
Therefore, if $\Hcts^{s}(\Gn, (E_n)_t/p)$ are all finite, the horizontal vanishing line implies the ending filtration has only finitely many elements, so $\pi_{t-s}\LKn(\S/p)$ is finite for all $m=t-s$.
We have deduced the argument into showing the continuous cohomology $\Hcts^q(P_n, (E_n)_t/p)$ is finite.
$p$-adic Geometry
The Honda formal group law $\Gamma_n$ has the property $[p]_{\Gamma_n} T=T^{p^n}$.
Let $k$ be a perfect field of characteristice $p$ ($\Fpn$ is good enough for our argument), $n \leq m\geq 1$ $A_m:=k[[u_m, …, u_{n-1}]$, $x=u_m$, and $r=n-m$. Let $F$ denote the restriction of the universal deformation $\widehat{ \Gamma_n}$ of $\Gamma_n$ on $A_m$, that is the pushforward of $\widehat{ \Gamma_n}$ along $W(k)[u_1,…,u_{n-1}]\to A_m$. So $F$ is the universal deformation of $\Gamma_n$ on height $\geq m$.
Therefore, we have the following property regarding $F$ \[ [p]_FT= xT^{p^m}+…=Q(T^{p^m}) \] for some power series $Q$.
Let $B_m:= A_m[1/x]$, this is then the locus of the deformation of $\Gamma_n$ of height exactly $m$, as $x$ is invertible.
Let $g\in P_n$, there is a change of coordinate $\alpha_g: g^*F\to F$. Let $\chi_g =\alpha’_g(0)$, then $g(x)=\chi_g^{p^m-1}x$. Denote $\mathcal{L}_m=A_m \cdot u$ where $g(u)=\chi_g$, the invariant differential line bundle over $A_m$, an invertible line bundle. We have $L_m^{\otimes t}=(E_n)_{2t}$ as continuous $P_n$-module.
Let $\ell\geq 0$, the $p^\ell$-torsion locus of $F$ is described by \[ A_m[[T]]/([p^\ell]_F T) \]
Let $0\leq b\leq m\ell$, denote $D_m(\ell, b)$ the subalgebra of $A_m[[T]]/([p^\ell]_F T)$ generated by $T^{p^b}$. This describes image of $b$-fold Frobenius map ($a\mapsto a^p$ for $b$-times, which is defined as $k$ is characteristic $p$) of the $p^\ell$-torsion of $F$.
Definition 2.3
[OAI26], Definition 3.1 The basic coefficients are \[ \mathcal{L}_m^{\otimes t}\otimes_{A_m}\bigotimes_{i=1}^s D_m(\ell_i, b_i) \] for $s\geq 0$, $\ell_i\geq 0$ and $0\leq b_i\leq m\ell_i$, and $t\in \mathbb{Z}$.
The allowed coefficient is the smallest class of $A_m$-modules $\mathcal{C}_m$ containing the basic coefficients closed under finite direct sum, tensor product, taking equivariant direct summand, and finite equivariant filtrations whose successive quotients belong to the class
We can think of $\bigotimes_{i=1}^s D_m(\ell_i, b_i)$ as the geometric intersections of the $b_i$-fold Frobenius points of $p^{\ell_i}$-torsions, and tensoring with $\mathcal{L}_m^{\otimes t}$ takes the invariant differentials over the geometric intersection.
Theorem 2.4 (Theorem I)
For every $1\leq m\leq n$, for every $M\in \mathcal{C}_m$ and every $i\geq 0$, \[ \Hcts^i(P_n, M) \]is finite.
proof: This argument follows from reverse induction on $m$, going from $n$ to $1$. However, we need to gather some necessary results to deduce this. We leave the proof of this theorem for later.
Q.E.D
We see that Theorem I implies the finite generatedness of $\pi_t \LKn \S$.