The Bennequin-sharpness characterization of strongly quasipositive links

A link $L$ is Bennequin-sharp when $-\chi(L)=\overline{sl}(L)$, and it is strongly quasipositive when it is represented as the closure of a product of positive band generators.

Strong quasipositivity conjecture. A link $L$ is strongly quasipositive if and only if it is Bennequin-sharp.

Strongly quasipositive links are known to be Bennequin-sharp; the conjecture asserts the converse and would characterize Bennequin-sharpness in terms of braid positivity.