The genus-independence conjecture for the monopole Floer Legendrian invariant

Let $\mathcal{K}\subset(Y,\xi)$ be a Legendrian knot, and let $\ell_g(\mathcal{K})\in KHM(-Y,K)\otimes\mathcal{R}$ denote the invariant constructed using a closure of genus $g$, for $g\geq2$. Genus-independence conjecture. The elements $\ell_g(\mathcal{K})$, $g\geq2$, are all equal as elements of $KHM(-Y,K)\otimes\mathcal{R}$ up to automorphism. This would show that the construction is independent of the auxiliary closure genus, up to the natural ambiguity from automorphisms. The source gives no resolution of this conjecture.