The Alexander polynomial coefficient inequality for rational cuspidal curves

Let $C$ be a rational cuspidal curve of degree $d$ with critical points $z_1,\ldots,z_n$. Let $K_1,\ldots,K_n$ be the corresponding links of singular points, with Alexander polynomials $\Delta_1,\ldots,\Delta_n$, and let $G_1,\ldots,G_n$ be their gap sequences. Set

$$g=|G_1|+|G_2|+\ldots+|G_n|$$

so that $g$ is the genus of $K$. Write

$$\Delta(t)=1+(t-1)g+(t-1)^2\sum_{j=0}^{2g-2}k_l$$

and let $I=I_1\diamond I_2\diamond\ldots\diamond I_n$. The Alexander polynomial coefficient inequality. For every $j=0,\ldots,d-3$,

$$k_{d(d-j-3)}\leq I\bigl(d(d-j-3)+1\bigr),$$

with equality for $n=1$. This conjecture concerns restrictions on the Alexander polynomials and gap sequences of rational cuspidal curves; the equality case for one critical point was verified by Borodzik and Livingston, while the general case remains open in the supplied source.