Representation-independent Toda Frobenius manifold mirror

Let $\mathfrak{g}$ be a simple Lie algebra, let $\bar i$ be a marked simple root, and let $X_{\mathfrak g,\bar i}^{\rm Toda}=\mathbb{C}^\star\times(\mathbb{C}^\star)^{r_{\mathfrak g}}$ carry the Landau–Ginzburg residue pairing and product defined by the spectral curve $\Gamma_u^{(\bar i)}$. Mirror symmetry for DZ Frobenius manifolds. These formulas define a semisimple, conformal Frobenius manifold $(X_{\mathfrak g,\bar i}^{\rm Toda},\eta,e,E,\star)$ independent of the irreducible representation $\rho$ and non-trivial Weyl orbit $W_{\bar k}^{\rho}$; the identity and Euler fields are

$$e=u_0^{-1}\partial_{u_{\bar i}},\qquad E=u_0\partial_{u_0},$$

and

$$X_{\mathfrak g,\bar i}^{\rm Toda}\simeq\widetilde X_{\mathfrak g,\bar i}.$$

Thus the Toda Landau–Ginzburg model is conjectured to realize the corresponding Frobenius manifold, independently of the auxiliary representation-theoretic choices.