The colored Alexander polynomial formula for hook partitions

Let $\mathcal{K}$ be a knot, let $A(\mathcal{K};q)$ denote its classical Alexander polynomial, and let $A_{\lambda}(\mathcal{K};q)$ denote the colored Alexander polynomial associated with a partition $\lambda$. A partition is a hook partition if it has the form $(m|n)$, equivalently $(m+1,1^n)$. The hook-partition Alexander conjecture. For every hook partition $\lambda$,

$$A_{\lambda}(\mathcal{K};q)=A(\mathcal{K};q^{|\lambda|}).$$

The formula is known to hold for torus knots, while it fails for general non-hook partitions such as $(2,2)$; its validity for arbitrary knots and hook partitions is the remaining conjectural assertion.