The $(2,3,7)$ hypergeometric-family Hodge-structure conjecture

Let $E_7$ be an elliptic curve with complex multiplication by $\mathbb{Q}(\sqrt{-21})$, and let

$$X_{7,t}: y^{84}=x^{13}(x-1)^{43}(x-t)^{29}$$

be a fiber of the hypergeometric family. The $(2,3,7)$ Hodge-structure conjecture. For every fiber $X_{7,t}$, the Hodge structure

$$\left(H^1(X_{7,t},\mathbb{Q})^{\otimes 3}\otimes_{\mathbb{Q}}H^1(E_7,\mathbb{Q})^{\otimes 2}\right)(2)$$

has a summand isomorphic to $H^1(C_{7,t},\mathbb{Q})$. The source proposes this as a conjectural realization of the relevant Hodge structure; no resolution is given.