Toric splitting conjecture for Calabi–Yau threefold zeta functions

Let $X$ be a Calabi–Yau threefold with $h^{1,1}(X)\neq h^{1,1}_{\mathrm{toric}}(X)$, and set

$$\tilde{h}=h^{1,1}(X)-h^{1,1}_{\mathrm{toric}}(X).$$

Let $\chi$ be a character depending on the defining equation of a singular locus. Toric splitting conjecture. The denominator of the zeta function takes the form

$$(1-t)(1-pt)^{h^{1,1}_{\mathrm{toric}}}(1-p^2t)^{h^{1,1}_{\mathrm{toric}}}(1-\chi pt)^{\tilde{h}}(1-\chi p^2t)^{\tilde{h}}(1-p^3t).$$

At this same singular locus, it takes the form

$$(1-t)(1-pt)^{h^{1,1}_{\mathrm{toric}}}(1-p^2t)^{h^{1,1}_{\mathrm{toric}}}(1-p^3t).$$

The conjecture is motivated by the mirror octic and by the relation between toric and non-toric divisors. Its status is explicitly open in the supplied metadata.