Let $A$ and $B$ be $n\times n$ matrices and let $p,q\in(1,\infty)$ such that $\frac{1}{p}+\frac{1}{q}=1$. Hölder's inequality for Schatten norms states that $$ \lvert \operatorname{Tr}(AB)\rvert \leq \lVert A\rVert_p \,\lVert B\rVert_q. $$ (See this paper for example.) Moreover, equality holds if and only if $\lvert A\rvert ^p$ is a multiple of $\lvert B\rvert^q$. This is similar to Hölder-type inequalities in other vector spaces. For $L^p$-norms, there is also a reverse Hölder's inequality (see Wikipedia for example).
Is there a reverse Hölder's inequality for matrices? That is, if $B$ is invertible, I'd like to know under what circumstances the following holds $$ \lvert \operatorname{Tr}(AB)\rvert \geq \lVert A^{1/p}\rVert_1^p \,\lVert B^{-1/q}\rVert_1^{-q}. \tag{1} $$ The proof given on the Wikipedia page doesn't hold for the case of matrices, since matrices don't necessarily commute. However, the inequality in (1) does hold if $A$ and $B$ commute.