The degree-one map conjecture for Heegaard genus

Let $M$ and $N$ be closed orientable 3-manifolds. A degree-one map is a map $f\colon M\to N$ of degree one, and $g(M)$ denotes the Heegaard genus of $M$. The degree-one map conjecture. If there is a degree-one map

$$f\colon M\to N,$$

then

$$g(M)\ge g(N).$$

This is an old and difficult question in 3-manifold topology. It would imply the Poincaré Conjecture, although the general statement remains open.