The rationality conjecture for the polarized index

Let $V$ be a smooth quasi-projective algebraic variety over ${\bf C}$ with an ample metrized invertible sheaf ${\cal L}$. Let $\alpha_{\cal L}(V)$ denote its ${\cal L}$-index, defined using a resolution and the cone of effective divisors.

Rationality conjecture. If

$$\alpha_{\cal L}(V)>0,$$

then $\alpha_{\cal L}(V)$ is rational.

The source says this follows from the Minimal Model Program and therefore holds in dimension at most $3$.