Prove the matrix Holder inequality $|{\rm Tr}(DAB)|\le {\rm Tr}(D|A|^p)^{1/p} {\rm Tr}(D|B|^q)^{1/q}$

Question

I am interested in the following version of the Holder inequality. Let $D \in M_n(\mathbb{C})$ be a positive semi-definite matrix of trace $1$ and $A, B \in M_n(\mathbb{C}).$ Does it follow that $$ |\mbox{Tr } DAB| \leq (\mbox{Tr } D|A|^p)^{1/p} (\mbox{Tr } D|B|^q)^{1/q}, $$ where $1/p + 1/q = 1$?

I found the case $D = I$ with proof (for the tracial state) in the literature but nothing more.