Fermat and related curve Rogers–Zudilin type conjectures

Let $F_{N,M}$ be the Fermat curve defined by

$$z^N+w^M=1,$$

and let $F^*_{2,4}$ be the curve $z^2=w^4+1$. Let $\sigma=\sigma_1$, so $\sigma(t)=t^p$. Fermat and related curve conjectures. There are rational numbers $C,C',C”\in\mathbb Q^\times$, independent of $p$, such that

$$(1-p\alpha_{F_{3,3},p}^{-1}){\mathscr F}_{\frac13,\frac13}^{(\sigma)}(1)=CL_p(F_{3,3},\omega^{-1},0),$$ $$(1-p\alpha_{F_{2,4},p}^{-1}){\mathscr F}_{\frac12,\frac14}^{(\sigma)}(1)=C'L_p(F_{2,4},\omega^{-1},0),$$ $$(1-p\alpha_{F^*_{2,4},p}^{-1}){\mathscr F}_{\frac14,\frac14}^{(\sigma)}(1)=C”L_p(F^*_{2,4},\omega^{-1},0).$$

These are the conjectures obtained from the regulator results for Fermat and related curves. The source provides no resolution status beyond their conjectural formulation.