Global functional equation conjecture for dualizing sheaves on singular curves

Assume $(*)$, let $X$ be a projective curve of arithmetic genus $g_a$, let $d\geq 0$, and let $\omega_X$ be a dualizing sheaf. Set

$$\mathcal{E}:=\omega_X^{\oplus d}.$$

Global functional equation conjecture. As rational functions in $t$,

$$Z_{\mathcal{E}}(t)=(\mathbb{L}^{d^2}t^{2d})^{g_a-1}Z_{\mathcal{E}}(\mathbb{L}^{-d}t^{-1})\in K_0(\operatorname{Var}_k)(t).$$

This is the global counterpart of the local functional equation and is part of the proposed extension from planar to Gorenstein singularities. No resolution is stated in the source.