Complexity conjecture for cohomology vanishing over finite $\Gamma$-semirings

Let $T$ be a finite $\Gamma$-semiring, and consider the problem of deciding whether

$$H^k_\Gamma(\operatorname{Spec}_\Gamma(T),\mathcal O_X)=0.$$

Complexity conjecture. This problem lies in $\mathbf{coNP}$, and becomes $\mathbf{P}$-complete for cyclic additive semirings. The claim gives a proposed complexity classification for computational $\Gamma$-cohomology; the source supplies no resolution or supporting result.