Nonlinear alternating Euler sums

Question

Let $H_{n}^{(s)}=\sum_{k=1}^{n}k^{-s}$ and $\overline{H}_{n}^{(s)}=\sum_{k=1}^{n}(-1)^{k+1}k^{-s}$. Flajolet - Salvy Theorem says that nonlinear Euler sum of the form $$ \sum_{n=1}^{\infty}\frac{H_{n}^{(p_{1})}H_{n}^{(p_{2})}\cdots H_{n}^{(p_{k})}}{n^{q}} $$ can be reduced to zeta values and sums of lower order when $p_{1}+p_{2}+\dots+p_{k}+q$ and $k$ are the same parity. My question is, if they exist a version of this Theorem about nonlinear alternating Euler sums, that is sums of the form $$ \sum_{n=1}^{\infty}\frac{H_{n}^{(p_{1})}H_{n}^{(p_{2})}\cdots H_{n}^{(p_{k})}\overline{H}_{n}^{(r_{1})}\overline{H}_{n}^{(r_{2})}\cdots \overline{H}_{n}^{(r_{l})}}{n^{q}}, $$ when $k+l\geq 2$?