Exceptional-factor conjecture for Calabi–Yau hypersurface zeta functions

Let a family of Calabi–Yau hypersurfaces have Hodge numbers $h^{2,1}$ and $h^{2,1}_{\mathrm{poly}}$, and let $R_{\mathrm{excep}}(t)$ denote a factor of the numerator of its zeta function. Exceptional-factor conjecture. If

$$h^{2,1}>h^{2,1}_{\mathrm{poly}},$$

then the numerator of the zeta function has a factor $R_{\mathrm{excep}}(t)$ independent of the parameters labelling the members of the family, with degree

$$\operatorname{deg}(R_{\mathrm{excep}})=2\acute{h},\qquad \acute{h}=h^{2,1}-h^{2,1}_{\mathrm{poly}}.$$

The conjecture is supported in the octic example and is consistent with the absence of such a factor for the quintic, where $h^{2,1}=h^{2,1}_{\mathrm{poly}}=101$.