p-adic L-function conjecture for the elliptic K3 families

Let $n\in\{3,4,6\}$ and suppose that ${\mathscr E}_{n,a}^-$ is a singular K3 surface over ${\mathbb Q}$. Let $A_{n,a}=\sum a_nq^n$ be the corresponding Hecke eigenform of weight $3$, and let $\alpha_p$ be the unit root of $T^2-a_pT+p^2$. Let $\sigma$ be the Frobenius $\sigma(t)=a^{1-p}t^p$.

Elliptic-family p-adic L-function conjecture. There is a constant $C_{n,a}\in {\mathbb Q}^{\times}$ independent of $p$ such that

$$L_p(A_{n,a},\omega_{\mathrm{Tei}}^{-1},0)=C_{n,a}(1-p^2\alpha_p^{-1}){\mathscr F}^{(\sigma)}_{\frac1n,\frac{n-1}{n},\frac12}(t)|_{t=a}.$$

This is obtained by combining the explicit syntomic-regulator computation for the elliptic K3 family with the $p$-adic Beilinson framework. The source gives no resolution of the assertion.