Batyrev's polynomial-growth conjecture for rational-curve components on weak Fano varieties

Let $X$ be a smooth projective weak Fano variety. For a numerical class $\alpha \in \operatorname{Nef}_{1}(X)_{\mathbb{Z}}$, let $h(\alpha)$ denote the number of components of $\operatorname{Mor}(\mathbb{P}^{1},X,\alpha)$ that generically parametrize free curves. Let $F(Y,f^*L)$ denote the relevant face associated to a face-contracting morphism $f:Y\to X$, and let $F$ be its image under $f_*$.

Batyrev's conjecture. Then $h(m\alpha)$, considered as a function of $m$, is bounded above by a polynomial $P(m)$ whose degree is the largest relative dimension of a map

$$f_*:F(Y,f^*L)\to F$$

where $f$ is a face-contracting morphism and $\alpha\in F$.

This refines the expected polynomial component bound by relating the degree to contracted faces. The source presents it as an expectation following examples in which face-contracting maps produce linear growth.