Local existence and uniqueness conjecture for higher-complex-structure connections

Let $U$ be a neighborhood of the zero-section in T^*\bm\hat{\mathcal{T}}^n, and let $[(\mu_k,t_k)]\in U$. A connection

$$C(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^*$$

is considered up to unitary gauge. Local existence and uniqueness conjecture. Given an element near the zero-section [(\mu_k,t_k)]\in U\subset T^*\bm\hat{\mathcal{T}}^n, there is a unique, up to unitary gauge, flat connection $C(\lambda)=\lambda\Phi+A+\lambda^{-1}\Phi^*$ satisfying: $\Phi$ is induced by the higher complex structure $[(\mu_2,\ldots,\mu_n)]$; $C(\lambda)$ satisfies the reality condition $-C(-1/\bar{\lambda})^*=C(\lambda)$; and $t_k=\operatorname{tr}\Phi_1^{k-1}A_1$. This is the precise local form of the preceding uniqueness claim, in the generalized Toda setting, and remains conjectural near the zero-section.