The pre-Tannakian and semisimplicity conjecture for symplectic interpolation categories

Fix a finite field $\mathbf{F}$ of odd cardinality $q$. For $n\geq 0$, let $H_n=\operatorname{\mathbf{Sp}}_{2n}(\mathbf{F})$, let $k_n$ be an algebraically closed field, and let $\mathcal{T}_n$ be the category of finite-dimensional representations of $H_n$ over $k_n$. Let $X_n=k_n[V_n]$ for $V_n=\mathbf{F}^{2n}$, choose an ultrafilter $\mathcal{F}$ on $\mathbf{N}$, and let $k$ be the characteristic-zero ultraproduct of the $k_n$. Let $\mathcal{T}$ be the tensor subcategory generated by $X=(X_n)_{n\geq 0}$ in the ultraproduct of the $\mathcal{T}_n$, and let $t\in k$ be the categorical dimension of $X$, represented by $(q^{2n})_{n\geq 1}$. The pre-Tannakian and semisimplicity conjecture. If $t\neq 0$, then $\mathcal{T}$ is pre-Tannakian and has enough projective objects. Moreover, if $t$ is not of the form $q^{2n}$ with $n\in\mathbf{N}$, then $\mathcal{T}$ is semisimple. The conjecture seeks a structural description of these interpolation categories, including their projective objects and the exceptional parameters at which semisimplicity can fail.