Congruence-simple multiplicatively idempotent semirings

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