The coefficient-sign conjecture for cable knots of the figure-eight knot

Let $\bm{4_1}$ denote the figure-eight knot, and consider the cable knots

$$K_r=C_{(r,2)}(\bm{4_1})\subseteq S^3,$$

where $r>8$ is odd and the Alexander polynomial is monic. Write the invariant as

$$F_{K_r}(x,q)=\frac12\sum_{\substack{m\ge1\m\ \operatorname{odd}}}(x^{m/2}-x^{-m/2})f_m(q).$$

Let $\{f_m(q)\}$ be partitioned into disjoint subsets $\{f_t^+(q)\}_{t\in I^+}$ and $\{f_w^-(q)\}_{w\in I^-}$. Coefficient-sign conjecture. The first subset consists of elements with all positive coefficients, the second of elements with all negative coefficients, and each $f_w^-(q)$ has coefficients agreeing up to sign with an element of the positive subset. This conjectures a uniform sign classification and coefficient correspondence for the BPS-series coefficients of this cable-knot family; the source gives no resolution status.