Classification conjecture for finite simple semirings with irreducible semimodules

Let $(R,\vee,\circ)$ be a finite simple additively idempotent semiring with an absorbing greatest element, and suppose it possesses a finite idempotent irreducible semimodule without property $(*)$. Classification conjecture. $(R,\vee,\circ)$ is isomorphic to a semiring in Corollary~. The conjecture is motivated by a computation of all finite simple additively idempotent semirings of cardinality at most $10$: every semiring in the specified class found computationally is isomorphic to one produced by the construction in the cited corollary. Its general validity remains open.