Generating the mapping class group of a punctured surface by involutions

Let ${\mathrm{\Sigma }}_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_{\mathit{\text{g,b}}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm Mod(\Sigma_{$\mathit{\text{g,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 $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_{\mathit{\text{g,b}}})$. Brendle and Farb [BF] gave an answer in the case of $g\ge 3,b=0$ and $g\ge 4,b=1$, by describing a generating set consisting of 6 involutions. Kassabov showed that for every $b$ $\rm Mod(\Sigma_{$\mathit{\text{g,b}}$})$canbegeneratedby4involutionsif$g\geq 8$, 5 involutions if $g\ge 6$ and 6 involutions if $g\ge 4$. We proved that for every $b$ $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_{\mathit{\text{g,b}}})$ can be generated by 4 involutions if $g\ge 7$ and 5 involutions if $g\ge 5$.