Classification conjecture for finite congruence-simple semirings without nontrivial square-zero elements

Let $S$ be a congruence-simple semiring with a multiplicatively absorbing element $w$, and suppose that

$$x^2\neq w$$

for every $x\in S\setminus\{w\}$. Classification conjecture. If $S$ is finite, then exactly one of the following cases occurs: $S\cong\mathbb{T}_4$ or $\mathbb{T}_8$; $S$ is a finite field; or

$$S\cong V(G),$$

where $G$ is a finite group. This conjecture is the consequence drawn in the paper from the cited structural conjecture together with its preceding results; the supplied text gives no resolution.