ScalingStacks

Conjecture 1 If two representations $J,\widetilde{J}$ of a surface $S$ have the property that

(a) diagrams $J_{0}$ and $\widetilde{J}_{0}$ are isomorphic,

(b) diagrams $J_{1}$ and $\widetilde{J}_{1}$ and isomorphic,

then the maps $\theta_{J}$ and $\theta_{\widetilde{J}}$ are equal, up to an overall minus sign, $\theta_{J}=\pm\theta_{\widetilde{J}}.$