Green–Griffiths conjecture for holomorphic maps from complex Euclidean spaces

Let $f:{\mathbb C}^{p}\rightarrow X$ be a holomorphic map into a compact complex manifold $X$ of general type, meaning that its canonical bundle $K_X$ is big. Green–Griffiths conjecture. The image of $f$ lies in a proper subvariety $Y_f$ of $X$. The equidimensional case $p=n$ has been settled by Kobayashi–Ochiai, while the cases $p=1$ and general $p$ remain open in the stated generality.