Toda mirror conjecture for polynomial orbifold projective lines

Let $\mathfrak{g}$ be simply laced and choose $\bar i$ by

$$\bar i=\begin{cases}i=1,\dots,n,&\mathfrak{g}=\mathrm{A}_n,\\ n-2,&\mathfrak{g}=\mathrm{D}_n,\\ 3,&\mathfrak{g}=\mathrm{E}_n.\end{cases}$$

Let $C_{\mathfrak g}$ be the polynomial $\mathbb{P}^1$-orbifold

$$C_{\mathfrak g}=\begin{cases}\mathbb{P}(\bar i,n-\bar i+1),&\mathfrak{g}=\mathrm{A}_n,\\ \mathbb{P}(2,2,n-2),&\mathfrak{g}=\mathrm{D}_n,\\ \mathbb{P}(2,3,n-3),&\mathfrak{g}=\mathrm{E}_n.\end{cases}$$

Orbifold mirror conjecture. With the notation of the preceding Toda mirror construction,

$$X_{\mathfrak g,\bar i}^{\rm Toda}\simeq QH_{\rm orb}(C_{\mathfrak g}).$$

This predicts that, for the indicated simply-laced types and marked nodes, the Toda Frobenius manifold agrees with the orbifold quantum cohomology of the corresponding polynomial projective-line orbifold. The source says that this would follow from the preceding mirror conjecture together with known results, but does not establish it here.