matrix inequality for Schatten norm

Question

Let $A_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r_i, i=1, \ldots, N$ be independent Rademacher random variables. The following inequality gives a bound on the expectation of the p-th Schatten norm: $$ E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{p}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{p}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{p}}^{p}\right\}. $$ Question: whether some similar bounds known for p-th moment of the fixed q-th Schatten norm, i.e. something like $$ E\left\|\sum_{i=1}^N r_i A_i\right\|_{S_{q}}^{p}\leq C\sqrt p \max\left\{\left\|\left(\sum_{i=1}^NA_iA_i^*\right)^{1/2}\right\|_{S_{q}}^{p}, \, \left\|\left(\sum_{i=1}^NA_i^*A_i\right)^{1/2}\right\|_{S_{q}}^{p}\right\}. $$