Pseudo $\ll$-transitivity for finite semigroups of order at most eight

Let $S$ be a semigroup, and let $\ll$ be its relation as above. The semigroup $S$ is pseudo $\ll$-transitive if, whenever $u,s,t\in S$ satisfy $u\lll st$, $u\not\ll st$, and $(u,st)$ is $n$-essential, there exist elements $v_1,\ldots,v_n$ such that

$$u\ll v_1\ll\cdots\ll v_n\ll st$$

and the conditions defining pseudo $\ll$-transitivity hold. Here $\lll$ denotes the smallest partially ordered set containing $\ll$.

Pseudo $\ll$-transitivity conjecture. If

$$|S|\leq 8,$$

then $S$ is pseudo $\ll$-transitive.

This conjecture was verified by a C# program for semigroups of order at most eight; no general proof or resolution beyond that computational verification is supplied here.