Stabilization conjecture for colored knot Floer homology

Let $L_m$ be the $m$-fold cable link of a link $L$, and let $\overline{\mathbf{s}}$ be a normalized Alexander degree. The maps

$$\phi_0:\mathcal{H\!F\! L}^{\mathrm{stab}}\left(L_m;\overline{\mathbf{s}}\right)\to \mathcal{H\!F\! L}^{\mathrm{stab}}\left(L_{m+1};\overline{\mathbf{s}}\right)$$

are the connecting maps in the directed system defining colored homology. Stabilization conjecture. For every normalized Alexander degree $\overline{\mathbf{s}}$, these maps are isomorphisms for sufficiently large $m$. Consequently,

$$\dim \mathcal{H}_{D,j}(L;\overline{\mathbf{s}})=\dim \mathcal{H\!F\! L}^{\mathrm{stab}}_j\left(L_m;\overline{\mathbf{s}}\right)$$

for $m\gg 0$. This would strengthen the proved upper bound by identifying the stable colored-homology dimension with the dimension of a sufficiently large stabilized cable group.