Genus-recovery conjecture for essential surfaces from the HOMFLY polynomial

Let $L \subset \Sigma \times I$ be a link, where $\Sigma$ is a surface. An essential surface for $L$ is a surface of smallest genus in which $L$ can be embedded in the thickened surface $\Sigma \times I$. Genus-recovery conjecture. The genus of an essential surface for $L$ can be recovered from its HOMFLY polynomial $P(L)$. This would extend the preceding result for non-split alternating links essential in $\Sigma$ with projections arising from embedded graphs, where the genus of $\Sigma$ is recoverable from the HOMFLY polynomial. The general case is left open.