Asai p-adic L-function interpolation conjecture

Let $\mathcal{F}$ be the Hilbert modular form under consideration, and suppose that $k>k'$. Let $j$ be an integer with $k'<j\leq k$. The primitive and imprimitive Asai $p$-adic $L$-functions are denoted by $L_{\mathfrak{p},\mathrm{Asai}}(\mathcal{F})$ and $L_{\mathfrak{p},\mathrm{Asai}}^{\operatorname{imp}}(\mathcal{F})$, respectively, while $L_{\mathrm{Asai}}(\mathcal{F},s)$ and $L_{\mathrm{Asai}}^{\operatorname{imp}}(\mathcal{F},s)$ denote their complex counterparts.

Asai $p$-adic $L$-function conjecture. Both $L_{\mathfrak{p},\mathrm{Asai}}(\mathcal{F})$ and $L_{\mathfrak{p},\mathrm{Asai}}^{\operatorname{imp}}(\mathcal{F})$ should be analytic at $\mathbf{j}=j$, and

$$L_{\mathfrak{p},\mathrm{Asai}}(\mathcal{F})(j)=0\Longleftrightarrow L_{\mathrm{Asai}}(\mathcal{F},1+j)=0,$$ $$L_{\mathfrak{p},\mathrm{Asai}}^{\operatorname{imp}}(\mathcal{F})(j)=0\Longleftrightarrow L_{\mathrm{Asai}}^{\operatorname{imp}}(\mathcal{F},1+j)=0.$$

This conjecture relates the zeros of the primitive and imprimitive Asai $p$-adic $L$-functions to those of the corresponding complex $L$-functions at critical integers. The supplied source gives no resolution status.