Pseudo-Noguchi finiteness conjecture I

Let $X$ be a projective integral variety over $\mathbb{C}$. A pseudo-Brody hyperbolic variety is a variety that is Brody hyperbolic outside a proper closed subset. Pseudo-Noguchi conjecture I. If $X$ is pseudo-Brody hyperbolic, then

$$\operatorname{Bir}_{\mathbb{C}}(X)$$

is finite. The corresponding finiteness theorem is known for Brody hyperbolic varieties, while the pseudo-Brody hyperbolic case is stated as open in the source.