Stošić's color-shift conjecture for rational two-component links

Let $L$ be a rational two-component link with components colored by a fixed color $\Lambda^j$ and a variable color $\Lambda^i$. Let $HP_j(L,\Lambda^i)\in \mathbb{N}[a^{\pm 1},t^{\pm 1},q^{-1}][[q]]$ be the Hilbert–Poincaré series of the $\Lambda^i$-reduced $(\Lambda^i,\Lambda^j)$-colored HOMFLY homology.

Changing color on unknot components only shifts $q$-grading. There exist rational functions $F_j(L)\in \mathbb{Z}[a^{\pm 1},s^{\pm 1},t^{\pm 1}](q)$ such that, for all $i\geq j$,

$$HP_j(L,\Lambda^i)=F_j(L)(a,s=q^{i-j},t,q),$$

after expanding the right-hand side into a power series in $q$.

The conjecture predicts a uniform rational-function description of these colored HOMFLY homologies as the color varies. It is attributed to Marko Stošić and is discussed in connection with rational knots and links; its resolution status is not established in the supplied text.