A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree

We show that for every smooth generic projective hypersurface $X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$ such that $\operatorname{codim}_X Y\ge 2$ and for every non constant holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$, provided $\deg X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.