Bandini–Valentino's Atkin–Lehner conjecture for Drinfeld modular forms at level T

Let $A$ be the coefficient ring and let $T$ be the prime defining the level. For weights $k,l$, write $S_{k,l}(\mathrm{GL}_2(A))$ for the space of Drinfeld cusp forms for $\mathrm{GL}_2(A)$, and $S_{k,l}(\Gamma_0(T))$ for the corresponding space at level $\Gamma_0(T)$. Let $S_{k,l}^{T-\mathrm{old}}(\Gamma_0(T))$ and $S_{k,l}^{T-\mathrm{new}}(\Gamma_0(T))$ denote the $T$-old and $T$-new subspaces, and let $T_T$ and $U_T$ be the relevant Hecke operators. Bandini–Valentino's conjecture. (i) The operator $T_T$ acting on $S_{k,l}(\mathrm{GL}_2(A))$ has trivial kernel, (ii)

$$S_{k,l}(\Gamma_0(T))=S_{k,l}^{T-\mathrm{old}}(\Gamma_0(T))\oplus S_{k,l}^{T-\mathrm{new}}(\Gamma_0(T)),$$

and (iii) $U_T$ is diagonalizable on $S_{k,l}(\Gamma_0(T))$. The conjecture would establish the expected oldform–newform decomposition and diagonalizability of the $U_T$-operator for Drinfeld modular forms at this prime level; the supplied text gives no resolution status.