p-adic L-function conjecture at the parameter a=1

Let $p\equiv1\pmod 4$ be a prime. Let $A=\eta^6(4z)=\sum a_nq^n$, and let $\alpha_p\in {\mathbb Z}_p$ be the root of $T^2-a_pT+p^2$ satisfying $\alpha_p\equiv a_p\pmod p$. Let $\sigma$ be the Frobenius $\sigma(t)=t^p$.

The $a=1$ p-adic L-function conjecture. There is a constant $C_1\in {\mathbb Q}^{\times}$ independent of $p$ such that

$$L_p(A,\omega_{\mathrm{Tei}}^{-1},0)=C_1(1-p^2\alpha_p^{-1}){\mathscr F}^{(\sigma)}_{\frac12,\frac12,\frac12}(t)|_{t=1}.$$

This is the proposed extension of the preceding conjecture to $a=1$; the source describes it as plausible in view of an earlier theorem, so its status remains open.