A Maeda-style conjecture for special eigenvalues of Drinfeld cusp forms

Let $A=\mathbb{F}_3[t]$, let $n\geq 2$, and set $k_n=1+3^n$. Let $\Sigma_n\subset(\mathbb{F}_3^\times)^n$ be the set of sign tuples with exactly $i$ entries equal to $-1$ for some $i$ satisfying $0\leq i\leq n$ and $i\equiv 1-n\pmod 3$. Let $S^0_{k_n,n}(\Gamma)$ and $S^0_{k_n,n-1}(\Gamma)$ denote the relevant double-cusp-form spaces, and let $P^0_{t,n,n}$ and $P^0_{t,n,n-1}$ be their corresponding Hecke polynomials. The special-eigenvalue conjecture. For all $n\geq 2$, the elements

$$\left\{t^{k_n/2}\sum_{i=0}^{n-1}c_{n,i}t^{-3^i}\mid (c_{n,i})_i\in\Sigma_n\right\}\subset A$$

are eigenvalues of the Hecke operator associated to $t$ acting on $S^0_{k_n,n}(\Gamma)$. Every remaining irreducible factor $Q_i$ of $P^0_{t,n,n}$ has degree greater than one and Galois group $S_{\deg Q_i}$, while the factorization of $P^0_{t,n,n-1}$ contains only such polynomials $Q_i$. This conjecture proposes a precise Maeda-style pattern for the Hecke factors of these Drinfeld cusp-form spaces; the numerical evidence described in the paper suggests the pattern, but its validity for all $n$ remains open.