p-adic Asai transfer

Let $K\mathrm{/}Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $G{L}_2\mathrm{/}K$, let $A{s}^{\pm }(\pi )$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $G{L}_4\mathrm{/}Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $G{L}_2\mathrm{/}K$ to the universal eigenvariety of $G{L}_4\mathrm{/}Q$, which at nice classical points interpolate this Asai transfer.