The tensor-subcategory conjecture for injective annihilator maps

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$, and let $\mathcal{C}$ be a tensor category of moderate growth over $\Bbbk$. Let $M^{\oplus}(\mathcal{C})$ be its split Grothendieck semiring and $\operatorname{TSpec}\Bbbk S_\infty$ the semiring of T-prime ideals. Let $\mathtt{Ver}_p$ be the Verlinde category.

Tensor-subcategory conjecture. If $\mathcal{C}$ is Frobenius exact and the annihilator map

$$\operatorname{TAnn}_{\mathcal{C}}:M^{\oplus}(\mathcal{C})\to\operatorname{TSpec}\Bbbk S_\infty$$

is injective, then $\mathcal{C}$ is a tensor subcategory of $\mathtt{Ver}_p$, and hence is incompressible.

This is presented as a more tractable special case of the preceding conjecture. The source does not report a resolution, while noting that the relevant characteristic-zero analogy does not directly prove the positive-characteristic statement for $p>3$.