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The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connectionsare known to be singular SYZ-mirror partners. We establish the equality ofHodge numbers of their intersection (stringy) cohomology. In rank two, thisanswers a question raised by Tam\'as Hausel in Remark 3.30 of "Global topologyof the Hitchin system".

Topological mirror symmetry for rank two character varieties of surface groups