Quantum cohomology invariance conjecture for projective toric bundles

Let $\mathbb P(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k))$ and $\mathbb P(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))$ denote the corresponding projective bundles over $\mathbb P^1$. Quantum cohomology invariance conjecture. For all $k\in\mathbb Z$,

$$QH^*\bigl(\mathbb P(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k))\bigr)\cong QH^*\bigl(\mathbb P(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))\bigr).$$

The source says that this conjecture is verified using the J-function and the ordinary mirror map, with an alternative derivation via connection matrices on the moduli space. Thus it is treated as solved.