Sub-irreducibility is equivalent to quotient-irreducibility for finite semirings

Let $(R,+,\cdot)$ be a finite, simple, additively idempotent semiring, and let $(M,+)$ be a finite idempotent $R$-semimodule. A semimodule is sub-irreducible if it is non-quasitrivial and has only id-quasitrivial proper subsemimodules; it is quotient-irreducible if it is non-quasitrivial and has only the trivial proper quotient semimodule. Sub-irreducibility conjecture. $(M,+)$ is sub-irreducible if and only if it is quotient-irreducible. The claim concerns the relationship between subsemimodules and quotient semimodules in finite simple additively idempotent semirings; the source reports computational evidence but gives no resolution.