Eventual stability conjecture for recurrence polynomials in the False Tate curve extension

For each integer $n\geq 1$, let $\left\{{\mathcal{R}}_p^{m,n}(T)\right\}_{m\geq 3}$ be the series of recurrence polynomials from the stated theorem. Eventual stability conjecture. For every $n\geq 1$, this series can be chosen to be eventually stable: for all sufficiently large $m$,

$${\mathcal{R}}_p^{m,n}={\mathcal{R}}_p^{m+1,n}={\mathcal{R}}_p^{m+2,n}=\cdots.$$

For $n=1$, eventual stability is already known, so the conjecture asks whether the analogous stabilization holds for every $n\geq 1$.