Conjecture on the quantum cohomology of a three-sphere-type Lagrangian

Let $L$ be the total space of the fibration defined by the theorem preceding the conjecture, with $k=1$, $p_1=4$, $p_2=11$, and $n=20$. Thus $L$ is the specific monotone Lagrangian embedded in $\mathbb{C}P^{19}$ by that theorem. A Lagrangian is wide if its quantum cohomology is isomorphic to its ordinary cohomology, and narrow if its quantum cohomology vanishes. The conjecture. The quantum cohomology of $L$ is nonzero but is not isomorphic to its ordinary cohomology:

$$QH^{*}(L,\mathbb{Z}_2[T,T^{-1}])\neq 0$$

and

$$QH^{*}(L,\mathbb{Z}_2[T,T^{-1}])\not\cong H^{*}(L,\mathbb{Z}_2[T,T^{-1}]).$$

Equivalently, $L$ is neither wide nor narrow. If true, this would provide an example outside the wide-or-narrow dichotomy; the source gives no resolution status.