The lattice-cohomological index conjecture for rational cuspidal plane curves

Let $L=S^3_{-d}(K)$ be the link of a superisolated surface singularity corresponding to a rational cuspidal projective plane curve of degree $d$. Let $\mathbb{H}^{\ast}_{\mathrm{can}}(L)$ denote its lattice cohomology in the canonical $\operatorname{Spin}^c$-structure. Lattice-cohomological index conjecture. One has

$$\operatorname{eu}\mathbb{H}^{\ast}_{\mathrm{can}}(L)\leq\frac{d(d-1)(d-2)}{6},$$

and equivalently,

$$\operatorname{eu}\mathbb{H}^{\ast}_{\mathrm{can}}(L)\leq\operatorname{eu}\mathbb{H}^{0}_{\mathrm{can}}(L).$$

This is the lattice-cohomological reformulation of the weaker index-theoretic conjecture. The corresponding equality for $\mathbb{H}^{0}_{\mathrm{can}}$ is proved in the paper, while the displayed inequality remains conjectural in general.