Fernández de Bobadilla–Luengo–Melle-Hernández–Némethi conjecture on Alexander polynomials of rational cuspidal curves

Let $C$ be a rational cuspidal curve of degree $d$ with critical points $z_1,\dots,z_n$. Let $K_1,\dots,K_n$ be the corresponding links of singular points, and let $\Delta_1,\dots,\Delta_n$ be their Alexander polynomials. Set $\Delta=\Delta_1cdotdotscdotDelta_n$ and expand it as

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

Fernández de Bobadilla–Luengo–Melle-Hernández–Némethi conjecture. For every $j=0,\dots,d-3$,

$$k_{d(d-j-3)}\leq \frac{(j+1)(j+2)}{2},$$

with equality when $n=1$. This conjecture gives constraints on the Alexander polynomials of the links of singular points of rational cuspidal curves. The source states that it was proposed in the cited work and verified for all known examples; the paper's theorem proves the one-singular-point case, while the general multiple-singularity statement is the relevant remaining conjectural form.