For a positive definite matrix $Q$ and any square matrices $A_i$, if $\sum_{i=1}^pA_i^TQA<Q$, can we get $\rho(\sum_{i=1}^pA_i^TA)<1$?

Question

I found that for a positive definite matrix $Q$ and a square matrix $A$, if $A^TQA<Q$, then $\rho(A)<1$. But what if $\sum_{i=1}^pA_i^TQA<Q$? can we get $\rho(\sum_{i=1}^pA_i^TA)<1$?