A conjectural Hilbert-modular construction for certain ${\mathbb Q}(\alpha)$-curves

Let $K=\mathbb Q(\sqrt D)$, let $I$ be a level, and let $f$ be a Hilbert cuspidal newform of level $I$ over $K$. Suppose there exist a quadratic field $\mathbb Q(\sqrt n)$ and a quadratic extension $L/K$ such that $e_{\mathfrak p}(f)$ is rational for primes $\mathfrak p$ split in $L/K$, and is $\sqrt n$ times a rational number for primes $\mathfrak p$ inert in $L/K$. Generalized Hamahata conjecture. There exist an elliptic curve $E/L$ isogenous to its Galois conjugate over $K$, a K3 surface $S$ defined over $\mathbb Q$ and $K$-isogenous to the Kummer surface of $E\times E^\sigma$, and a correspondence from $S$ to the Hilbert modular surface $H_{I,K}$, where $I$ is the conductor of $E$ viewed as an ${\mathcal O}_K$-ideal.

This is proposed as a generalization of Hamahata's conjecture to certain $\mathbb Q(\alpha)$-curves. The supplied text gives no evidence that it has been resolved.