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