The period-coordinates conjecture for the spectral curve

Let $\tilde{\Sigma}$ be the spectral curve associated with a point of T^*\bm\bm\bm\bm\bm\bm\bm\bm\hat{\mathcal{T}}^n, and let $\alpha$ be the Liouville form on $T^{*\mathbb{C}}\Sigma$, locally $\alpha=p\,dz+\bar p\,d\bar z$. Spectral period-coordinate conjecture. Restriction of $\alpha$ to $\tilde{\Sigma}$ gives a local isomorphism

T^*\bm\bm\bm\bm\bm\bm\bm\bm\hat{\mathcal{T}}^n\cong H^1(\tilde{\Sigma},\mathbb{C})/H^1(\Sigma,\mathbb{C}),

and some ratios of periods give a coordinate system on \bm\bm\bm\bm\bm\bm\bm\bm\hat{\mathcal{T}}^n. The source notes that the spectral curve is only Lagrangian modulo $t^2$, which is the principal obstruction to proving the conjecture.