Density-degree conjecture for products of hyperelliptic curves

Let $C$ and $D$ be hyperelliptic curves over $\mathbb{Q}$, and let $\delta(C\times D/\mathbb{Q})$ denote the density degree set of their product. Write $\min\delta(C\times D/\mathbb{Q})$ for its least element.

Density-degree conjecture. Most pairs of hyperelliptic curves $(C,D)$ of sufficiently large genera satisfy

$$\min\delta(C\times D/\mathbb{Q})=4.$$

This is motivated by the fact that products of hyperelliptic curves have degree of irrationality $4$ and by expectations concerning the scarcity of unexpected rational and quadratic points, but the source leaves the precise meaning of “most” and the required genus threshold informal.