Fubini–Study Chern number conjecture for singular K3 surfaces

Let $X\subseteq\mathbb{P}_\mathbb{C}^3$ be a possibly singular K3 surface, and let $X_s$ be its smooth locus with curvature form $J$ induced by the Fubini–Study metric on $\mathbb{P}_\mathbb{C}^3$. Suppose that the singularities of $X$ are isolated and of type $A_1$. Fubini–Study Chern number conjecture. Then

$$\int_{X\backslash \operatorname{Sing}X} c_2(J)-c_1(J)^2=24=\chi(\operatorname{K3}).$$

The conjecture is prompted by numerical Monte Carlo evaluations for the family studied in the paper, where the displayed identity appears for the singular examples. Its general validity for all possibly singular K3 surfaces satisfying the stated hypotheses is not established here.