The monodromy subcover conjecture for splitting Jacobians

Let $X$, $Y$, $\bar{Y}$, and $W$ be the curves in the diagram

$$X \xleftarrow{\pi} Y \xrightarrow{\alpha_Y} \bar{Y} \xrightarrow{\alpha_W} W.$$

For a suitable choice of a subcover $W$, the algebraic correspondence

$$X \xleftarrow{\pi} Y \xrightarrow{\alpha_Y} \bar{Y} \xrightarrow{\alpha_W} W$$

induces the isogeny in the paper's hiding elliptic curve construction; equivalently, the map $\pi_* \times (\alpha_W)_*(\alpha_Y)^*$ is an isogeny.

This conjecture gives a monodromy-based geometric construction for the elliptic factor used to split the Jacobian of a two-dimensional Jacobian. The source presents it as a conjecture suggested by Lombardo's work; its resolution is not specified here.