Generating the mapping class group of a punctured surface by involutions

Let Σg,b denote a closed orientable surface of genus g with b punctures and let Mod(Σg,b) denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, Mod(Σg,b) is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate Mod(Σg,b). Brendle and Farb [BF] gave an answer in the case of g ≥ 3, b = 0 and g ≥ 4, b = 1, by describing a generating set consisting of 6 involutions. Kassabov showed that for every b Mod(Σg,b) can be generated by 4 involutions if g ≥ 8, 5 involutions if g ≥ 6 and 6 involutions if g ≥ 4. We proved that for every b Mod(Σg,b) can be generated by 4 involutions if g ≥ 7 and 5 involutions if g ≥ 5.