On conjugacies between piecewise-smooth circle maps

Let fi, i=1,2, be piecewise C1 circle homeomorphisms with two break points, logDfi, i=1,2, are absolutely continuous on each continuity interval of Dfi and DlogDfi∈Lp for some p>1. Suppose, the jump ratios of f1 and f2 at their break points do not coincide but f1,f2 have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the map h conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dh(x)=0 almost everywhere with respect to Lebesgue measure.