A criterion for a degree-one holomorphic map to be a biholomorphism

Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied.