Finiteness conjecture for Euler characteristics of Stein fillings

Let $(M,\xi)$ be a contact $3$-manifold, and let $\chi(X)$ denote the Euler characteristic of a compact manifold $X$. Define

$$\mathcal{C}_{(M,\xi)}=\{\chi(S)\mid S\text{ is a Stein filling of }(M,\xi)\}.$$

Finiteness conjecture. The set $\mathcal{C}_{(M,\xi)}$ is finite.

This conjecture proposes a finiteness property for Stein fillings of a fixed contact $3$-manifold. The paper constructs contact $3$-manifolds admitting infinitely many pairwise non-diffeomorphic Stein fillings, while the finiteness of their Euler characteristics is left open.