Batyrev's bounded primitive collections conjecture for complete nonsingular fans

Let $\Sigma$ be an $n$-dimensional complete nonsingular fan, let $G(\Sigma)$ be the set of generators of its rays, and let $\ell$ be the Picard number of $\Sigma$. A primitive collection in $G(\Sigma)$ is a primitive collection of rays of $\Sigma$. Batyrev's bounded primitive collections conjecture. For any $n$-dimensional complete nonsingular fan $\Sigma$ with Picard number $\ell$, there exists a constant $N(\ell)$ depending only on $\ell$ such that the number of primitive collections in $G(\Sigma)$ is always not more than $N(\ell)$. The paper states that this conjecture follows from its finiteness theorem and that a proof is given in a corollary, so the conjecture is considered solved.