Congruence-simple multiplicatively idempotent semirings

Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $\mathrm{\mid }S\mathrm{\mid }=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $\mathrm{\mid }S\mathrm{\mid }=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.