GNSSS color-shifting conjecture for reduced colored HOMFLY homology

Let $\mathcal{L} = \mathcal{L}_1 \sqcup \dots \sqcup \mathcal{L}_d$ be a link with $\mathcal{L}_1$ an unknot and suppose $p \in \mathcal{L}_1$. For colors $k_1,\dots,k_d$, let $\mathcal{P}_{k_1,\dots,k_d}(A,Q,T)$ denote the graded dimension of reduced colored HOMFLY homology at $p$. GNSSS color-shifting conjecture. There exists a polynomial $f_{k_2,\dots,k_d}(A,Q,T,x)$, independent of $k_1$, such that

$$\mathcal{P}_{k_1,\dots,k_d}(A,Q,T)=f_{k_2,\dots,k_d}(A,Q,T,x=Q^{k_1})$$

for all $k_1\geq 0$, with $\mathcal{P}_{0,k_2,\dots,k_d}:=\dim(HHH(\mathcal{L}-\mathcal{L}_1))$. This color-shifting behavior is wide open in full generality, including for non-torus links; its decategorified version for colored HOMFLY polynomials was proved by Wedrich.