Conjecture on the quadratic conductor formula for quasi-homogeneous hypersurface singularities

Let ${\mathcal O}$ be a discrete valuation ring with residue field $k$ of characteristic different from two, quotient field $K$ and parameter $t$. Let $f:{\mathcal X}\to\operatorname{Spec}{\mathcal O}$ be a flat and proper morphism with ${\mathcal X}$ smooth over $k$ and with generic fiber $X_K$ smooth over $K$. Suppose that the special fiber $X_0$ has only finitely many singular points $p_1,\ldots,p_s$. Suppose in addition that each $(X_0,p_i)$ is, over the algebraic closure $\overline{k(p_i)}$, a quasi-homogeneous hypersurface singularity of degree $e_i$, with weights $a_*^{(i)}$, and with $\operatorname{lcm}(a_*^{(i)})=1$. Let $a^{(i)}=\prod_j a_j^{(i)}$. Finally, suppose that $e_i$, $a^{(i)}$, and $[k(p_i):k]$ are prime to $\operatorname{char} k$ if this is positive. Define

$$\Delta_t({\mathcal X}/{\mathcal O}):={\operatorname{sp}}_t(\chi_c(X_K/K))-\chi_c(X_0/k)\in {\operatorname{GW}}(k).$$

Conjecture on the quadratic conductor formula. Then

$$\Delta_t({\mathcal X}/{\mathcal O})=\sum_{i=1}^s {\operatorname{Tr}}_{k(p_i)/k}[\langle e_i\cdot a^{(i)}\rangle-\langle1\rangle+(-\langle e_i\rangle)^{{\operatorname{dim}}_{\mathcal O}{\mathcal X}}e_{p_i}(\Omega_{{\mathcal X}/k},dt)].$$

The formula is a quadratic refinement of the conductor formula for a degeneration with isolated quasi-homogeneous hypersurface singularities. The source notes that, in characteristic zero, it has been verified when each singularity can be resolved by a single weighted blow-up with smooth exceptional divisor; the general statement remains conjectural.