Stabilization conjecture for cable links with fixed remainder

Fix a knot $K$, an integer $n$, and a remainder $r$, and consider the cable links $K_{n,mn+r}$ as $m$ varies. Let $\widetilde{\mathbf{s}}$ be the normalized Alexander degree preserved by the connecting maps

$$\phi_0:\mathcal{H\!F\! L}^{\mathrm{stab}}\left(K_{n,mn+r};\widetilde{\mathbf{s}}\right)\to \mathcal{H\!F\! L}^{\mathrm{stab}}\left(K_{n,(m+1)n+r};\widetilde{\mathbf{s}}\right).$$

Cable stabilization conjecture. For a given normalized Alexander degree $\widetilde{\mathbf{s}}$, the dimension of $\mathcal{H\!F\! L}^{\mathrm{stab}}(K_{n,mn+r},\widetilde{\mathbf{s}})$ stabilizes for sufficiently large $m$, and the maps $\phi_0$ are isomorphisms for sufficiently large $m$. This would imply that the colimit of the directed system for the cable family with fixed remainder $r$ is well defined.