Yui's modularity conjecture for rigid Calabi–Yau threefolds

Let $X$ be a rigid Calabi–Yau threefold of CM-type defined over a number field $F$. Its intermediate Jacobian $J^2(X)$ is an elliptic curve with complex multiplication by an imaginary quadratic field $K$, and has a model defined over $F$. Let $\chi$ be a Hecke character associated to $J^2(X)$ and suppose that

$$L(J^2(X),s)=\begin{cases}L(\chi,s)L(\overline{\chi},s)&\text{if }K\subset F,\\L(\chi,s)&\text{otherwise.}\end{cases}$$

Yui's conjecture. Then

$$L(X,s)=\begin{cases}L(\chi^3,s)L(\overline{\chi}^3,s)&\text{if }K\subset F,\\L(\chi^3,s)&\text{otherwise,}\end{cases}$$

and consequently $X$ is modular. This conjecture relates the arithmetic of a rigid Calabi–Yau threefold of CM-type to that of its intermediate Jacobian; the source presents it as the precise conjecture motivating the work, and its resolution is not established in the supplied material.