Parity conjecture for quantum cohomology of Hirzebruch surfaces

Let $F_n=\mathbb P(\mathcal O\oplus\mathcal O(-n))$ be the Hirzebruch surface over $\mathbb P^1$, in the range $n\geq 3$ considered by the source. Parity conjecture. There are two isomorphism types of $QH^*(F_n)$, according to whether $n$ is odd or even. This is proposed after computations using Birkhoff factorization and a generalized mirror transformation; the source gives no resolution evidence, so the conjecture remains open.