Finite presentation of end terms of almost split sequences

Let $0\to X\to Y\to Z\to 0$ be an almost split sequence in a module category.

Finite-presentation conjecture. The module $Z$ is finitely presented.

This conjecture asserts that almost split sequences cannot have a non-finitely presented right-hand term. The paper notes that very little is known about non-existence of almost split sequences and gives the example that the category of modules over $\mathbb Z$ has no almost split sequence ending in $\mathbb Q$.