Dehn twist–length operator conjecture for quantized Teichmüller space

Let $\mathfrak{S}$ be a triangulable marked surface and let $\Delta$ be its ideal triangulation. Let $\gamma$ be an essential non-peripheral simple loop, let $D_{[\gamma]} \in \operatorname{MCG}(\mathfrak{S})$ be the Dehn twist about $\gamma$, let $\rho^\hbar_{\Delta;\operatorname{MCG}}$ be the projective unitary mapping-class-group representation on $\mathscr{H}_\Delta$, and let $\mathbf{l}^\hbar_{[\gamma],\Delta}$ be the quantized length operator. Dehn twist–length operator conjecture. Up to a constant, one has

$$\rho^\hbar_{\Delta;\operatorname{MCG}}(D_{[\gamma]}) = \exp\left({\rm i}\frac{(\mathbf{l}^\hbar_{[\gamma],\Delta})^2}{8\pi\hbar}\right).$$

This conjecture would resolve the Dehn-twist ambiguity in the mapping-class-group action associated with cutting along $\gamma$ and would connect the mapping class group representation with the quantized geodesic length; its status is not resolved in the supplied text.