The annihilator-map characterization of Verlinde categories

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, $M(\mathcal{C})$ its Grothendieck ring, and let $\operatorname{TSpec}\Bbbk S_\infty$ and $\operatorname{TSys}\Bbbk S_\infty$ denote the semirings of T-prime ideals and T-systems, respectively. Set $\mathtt{Ver}_{p^\infty}=\bigcup_{n\geq 1}\mathtt{Ver}_{p^n}$.

Annihilator-map conjecture. The annihilator map

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

is injective if and only if $\mathcal{C}$ is incompressible, and is a semiring isomorphism if and only if $\mathcal{C}=\mathtt{Ver}_{p^\infty}$. Moreover,

$$\operatorname{TSys}:M(\mathtt{Ver}_{p^\infty})\to\operatorname{TSys}\Bbbk S_\infty$$

is a semiring isomorphism.

The conjecture strengthens the cited BEO conjecture and is intended to characterize incompressibility and the distinguished role of $\mathtt{Ver}_{p^\infty}$ through annihilator data. The source states that it is currently out of reach, although it gives supporting consequences and evidence.