Conjectures on Hecke operators, diagonalizability, and oldforms for Drinfeld cusp forms

Let $q$ be the cardinality of the coefficient field, let $t$ be the level-one prime, and let $S^1_{k,m}(GL_2(A))$ and $S^1_{k,m}(\Gamma_0(t))$ denote the corresponding spaces of Drinfeld cusp forms of weight $k$ and type $m$. Write $\mathbf{T}_t$ for the Hecke operator at $t$, $\mathbf{U}_t$ for the level-$t$ operator, and $S^{1,old}_{k,m}(\Gamma_0(t))$ and $S^{1,new}_{k,m}(\Gamma_0(t))$ for the oldform and newform subspaces, with $\delta$ the degeneracy-map construction and $Tr,Tr'$ the trace and twisted-trace maps.

The oldform and Hecke conjectures. The following assertions should hold:

$$\operatorname{Ker}(\mathbf{T}_t)=0;$$

$\mathbf{U}_t$ is diagonalizable when $q$ is odd and, when $q$ is even, it is diagonalizable if and only if the dimension of $S^{1,new}_{k,m}(\Gamma_0(t))$ is $1$; and

$$S^1_{k,m}(\Gamma_0(t))=S^{1,old}_{k,m}(\Gamma_0(t))\oplus S^{1,new}_{k,m}(\Gamma_0(t))=\operatorname{Im}(\delta)\oplus(\operatorname{Ker}(Tr)\cap\operatorname{Ker}(Tr')).$$

These conjectures are motivated by numerical data and comparison with the classical case. Non-diagonalizability is known in even characteristic and appears related to the antidiagonal action of $\mathbf{U}_t$ on newforms; the proposed decomposition and injectivity remain unresolved in the source.