The MRC-foliation characterization for varieties with nef anti-canonical bundle

Let $X$ be a smooth $n$-dimensional projective variety with nef anti-canonical bundle, and let $\mathcal{F}$ be a foliation on $X$. For some ample divisors $H_1,\ldots,H_{n-1}$ on $X$, set

$$\alpha=H_1\cdots H_{n-1}.$$

Here $\mu_{\alpha}^{\min}(\mathcal{F})$ denotes the minimal slope of $\mathcal{F}$ with respect to $\alpha$, and $K_{\mathcal{F}}\equiv K_X$ denotes numerical equivalence of canonical classes. MRC-foliation characterization. The foliation $\mathcal{F}$ is induced by the maximal rationally connected (MRC) fibration of $X$ if and only if

$$\mu_{\alpha}^{\min}(\mathcal{F})>0$$

and

$$K_{\mathcal{F}}\equiv K_X.$$

This conjecture proposes a numerical characterization of the foliation induced by the MRC fibration when the anti-canonical bundle of $X$ is nef. The preceding results establish related statements for regular foliations and morphisms under stronger assumptions, while the claimed equivalence is not resolved here.