Miyanishi's conjecture on endomorphisms of varieties

Let $X$ be a $\mathbb{C}$-variety and let $\phi:X\to X$ be an endomorphism that is injective outside a closed subset of codimension at least $2$ in $X$.

Miyanishi's conjecture. Then $\phi$ is an isomorphism.

This conjecture concerns when a nearly injective endomorphism of a variety must be an automorphism. It is solved: the stated form follows from the paper's result, while another part of Miyanishi's conjecture was previously proved by Das in 2022 using ideas of Kaliman.