Miyanishi's conjecture on endomorphisms injective outside a closed subvariety

Let $X$ be an algebraic variety over an algebraically closed field of characteristic zero, and let $\phi:X \longrightarrow X$ be an endomorphism. Let $Y$ be a proper closed subvariety of $X$ such that the restriction of $\phi$ to $X \setminus Y$ is injective. Suppose furthermore that either $\phi$ is quasi-finite or $Y$ has codimension at least $2$ in $X$. Miyanishi's conjecture. Then $\phi$ is an automorphism. The conjecture concerns conditions under which an endomorphism of an algebraic variety must be an automorphism. The paper's abstract states that both cases are proved: quasi-finite endomorphisms injective outside a closed subvariety, and endomorphisms of complex algebraic varieties injective outside a subvariety of codimension at least $2$.