Tsaknias's local-type conjecture for non-CM newform Galois orbits

Let $p$ be a prime and let $n\geq 1$. Consider non-CM newform Galois orbits of level $p^n$, weight $k$, and trivial Nebentypus. For an inertial local type $\tau$ of conductor $p^n$, write $[\tau]$ for its Galois orbit, and let $\lambda_{AL}$ denote an Atkin–Lehner eigenvalue compatible with $\tau$. Let $\operatorname{LO}(p^n)$ be the number of possible pairs $([\tau],\lambda_{AL})$. Tsaknias's local-type conjecture. The number of non-CM newform Galois orbits is eventually independent of the weight $k$ and equal to

$$\operatorname{LO}(p^n).$$

This conjecture proposes that local inertial types, together with compatible Atkin–Lehner eigenvalues, determine the eventual number of non-CM Galois orbits at prime-power level. It extends the proposed generalized Maeda conjecture to prime-power levels; the supplied context reports computational motivation but does not establish the claim.