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The short-distance expansion of the tau function of the radialsine-Gordon/Painlev\'e III equation is given by a convergent series whichinvolves irregular $c=1$ conformal blocks and possesses certain periodicityproperties with respect to monodromy data. The long-distance irregularexpansion exhibits a similar periodicity with respect to a different pair ofcoordinates on the monodromy manifold. This observation is used to conjecturean exact expression for the connection constant providing relativenormalization of the two series. Up to an elementary prefactor, it is given bythe generating function of the canonical transformation between the two sets ofcoordinates.

Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks