Diagonal-action mirror symmetry equivalence for local curves

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the diagonal torus action $\lambda_1=\lambda_2$, and let $\mathbb{P}(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k))$ be the associated projective bundle. Diagonal-action equivalence conjecture. The equivariant mirror-symmetry computation on $X_k$ with the diagonal torus action is the same as that of

$$\mathbb{P}(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k)),$$

so their mirror maps and Gromov–Witten invariants are equal. This phenomenon contrasts with the simpler Calabi–Yau structure of the antidiagonal action. The source gives no resolution evidence for this parsed conjecture.