Antidiagonal mirror symmetry formulas for local curves

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the antidiagonal torus action $(\lambda,-\lambda)$ on the bundle, and let $t_k$ denote the Kähler class of the zero section. Write $q$ for the exponentiated $B$-model coordinate, $Y^k_{qqq}$ for the rational $B$-model Yukawa coupling, and $\theta=q\frac{d}{dq}$. Antidiagonal mirror symmetry conjecture. The mirror map and rational $B$-model Yukawa coupling are given by

$$q\frac{dt_k}{dq}=\frac{1+(-1)^{k+1}(k+1)^2q}{1+(-1)^{k+1}q},$$ $$Y^k_{qqq}=\frac{-1}{k(k+2)}\left(q\frac{dt_k}{dq}\right)^2.$$

In particular, the Picard–Fuchs equation describing mirror symmetry for $X_k$ is

$$\partial_{t_1}\left(\frac{1}{Y^k_{t_1t_1t_1}}\right)\partial_{t_1}^2=\theta^2\left(q\frac{dt_k}{dq}\right)^{-1}\theta.$$

These formulas express the especially simple structure found for the equivariantly Calabi–Yau antidiagonal action and determine the corresponding rational Yukawa coupling and mirror differential equation. The source presents them as its main results, but supplies no resolution evidence for the conjectural status of the parsed statement.