Poonen's finiteness conjecture for modular curves of fixed genus

Let $X_1(N)$ be the usual modular curve over $\mathbb{Q}$, and call a curve $X$ over $\mathbb{Q}$ modular if there is a nonconstant morphism $X_1(N)\to X$ over $\mathbb{Q}$ for some $N$. For each integer $g\geq 2$, consider modular curves $X$ of genus $g$. Poonen's finiteness conjecture. For each $g\geq 2$, the set of modular curves over $\mathbb{Q}$ of genus $g$ is finite. The paper contrasts this with the genus-$0$ and genus-$1$ cases, where infinitely many modular curves occur; the conjecture concerns finiteness in every fixed genus at least $2$, and no resolution is given here.