Colored HOMFLY-PT homology limit conjecture

Let $B^{(m)}$ be the $m$-colored version of a braid diagram $B$, and let $H_N(B^{(m)})$ denote its colored $athfrak{sl}(N)$ link homology. Denote by $H$ the colored HOMFLY-PT link homology, with an appropriate normalization. The quantities $\min\deg_q$ and $\max\deg_q$ are the minimum and maximum $q$-gradings, respectively. Then colored HOMFLY-PT homology limit conjecture.

$$\lim_{N\rightarrow\infty}\frac{\min\deg_q H_N(B^{(m)})}{N-m}=\min\deg_x H(B^{(m)}),$$ $$\lim_{N\rightarrow\infty}\frac{\max\deg_q H_N(B^{(m)})}{N-m}=\max\deg_x H(B^{(m)}).$$

Here $\deg_x$ is the degree from the $x$-grading corresponding to the framing variable $x$ of the colored HOMFLY-PT link polynomial. In particular, if $w$ is the writhe and $b$ is the number of strands of $B$, then

$$w-b\leq\frac{\min\deg_x H(B^{(m)})}{m}\leq\frac{\max\deg_x H(B^{(m)})}{m}\leq w+b.$$

The conjecture is motivated by the expected generalization of Rasmussen's spectral sequence relating colored HOMFLY-PT link homology to colored $\mathfrak{sl}(N)$ link homology. A direct proof of the resulting Morton-Franks-Williams-type inequality was suggested as possible, but the claim remains unresolved in the source.