The Oda-Hamahata conjecture for real quadratic elliptic curves

Let $K$ be a real quadratic field, let $I$ be an ideal of ${\mathcal O}_K$, and let $H_{I,K}$ be the Baily–Borel compactification of the Hilbert modular surface of level $I$. For an elliptic curve $E$ over $K$ with conductor $I$, write $E^\sigma$ for its Galois conjugate and set

$$K_E=(E\times E^\sigma)/\pm 1.$$

Let $\operatorname{H}^2_{\mathrm{prim}}$ denote the primitive part of second cohomology. Oda-Hamahata conjecture. There is a correspondence between $H_{I,K}$ and $K_E$ that induces an injection

$$\operatorname{H}^2_{\mathrm{prim}}(K_E)\hookrightarrow \operatorname{H}^2_{\mathrm{prim}}(H_{I,K}).$$

The conjecture is inspired by Oda's work and is described as an important special case of the Tate conjecture. The supplied text gives no evidence that it has been resolved.