Gromov's conjecture on rational curves and mean dimension

Let $X\subset \mathbb{C} P^N$ be a projective manifold, and let $\mathcal{M}(X)$ be the space of Brody curves in $X$. Gromov's conjecture.

$$\dim(\mathcal{M}(X):\mathbb{C})>0 \quad\text{if and only if}\quad X\text{ contains a rational curve}.$$

The conjecture relates positivity of the mean dimension of the Brody-curve space to the existence of rational curves on the target. It is attributed in the source to Gromov; no resolution is supplied in the excerpt.