Bounds for the regularity of local cohomology of bigraded modules

Let $M$ be a finitely generated bigraded module over the standard bigraded polynomial ring $S=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m,y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_n]$, and let $Q=(y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_n)$. The local cohomology modules $H_Q^k(M)$ are naturally bigraded, and the components $H_Q^k(M{)}_j={\text{Dirsum}}_i{H}_Q^k(M{)}_{(i,j)}$ are finitely generated graded $K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_m]$-modules. In this paper we study the regularity of $H_Q^k(M{)}_j$, and show in several cases that $\text{reg}{H}_Q^k(M{)}_j$ is linearly bounded as a function of $j$.