Finiteness conjecture for degree-one maps to four-manifolds with finite fundamental group

Let $X$ and $Y$ be closed orientable $4$-manifolds. Say that $X$ 1-dominates $Y$ if there is a continuous map $f:X\to Y$ of degree one.

Finiteness conjecture. Every closed orientable $4$-manifold $X$ 1-dominates at most finitely many closed orientable $4$-manifolds $Y$ with finite $\pi_1(Y)$, up to homeomorphism.

The paper establishes this finiteness when the fundamental group is finite abelian and reduces the general finite-group case to a proposition stated immediately afterward. The supplied text does not establish the conjecture in full.