The order-four subsemigroup conjecture for finite non-adequate amiable semigroups

Let $S$ be a finite amiable semigroup, meaning that each of its Green's starred $\mathcal{L}^*$- and $\mathcal{R}^*$-classes contains exactly one idempotent. Let $M$ be the order-four amiable semigroup constructed above, with underlying set $\{a,b,c,d\}$ and multiplication determined by that construction.

Subsemigroup conjecture. If $S$ is not adequate, then $S$ contains a subsemigroup isomorphic to $M$.

A computer search found this property for all such semigroups of order at most $37$. The conjecture proposes that the unique order-four amiable semigroup that is not adequate is contained in every finite non-adequate amiable semigroup; no resolution is given here.