Determinant vanishing conjecture for finite semigroups in ECOM

Let $S$ be the finite semigroup under consideration, let $u,u'\in S$, let $u^{{+}_r}$ and ${u'}^{{+}_l}$ denote their associated right and left idempotents, let $\boldsymbol{*}$ denote the operation defined in the paper, and let $\theta_S(X)$ be the determinant polynomial associated with $S$. Determinant vanishing conjecture. If there exist elements $u$ and $u'$ such that

$$u^{{+}_r}\neq {u'}^{{+}_l}\quad\text{and}\quad u\boldsymbol{*}u'\neq 0,$$

then

$$\theta_S(X)=0.$$

This conjecture concerns the determinant of a finite semigroup in the pseudovariety ECOM; proving the stated condition forces the determinant to vanish would identify a structural obstruction to a nonzero determinant. The supplied text does not indicate whether the conjecture has been resolved.