  This note proves the following inequality: if $n=3k$ for some positive
integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$,
\begin{equation}
\frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\| \geq
\frac{(n-3)!}{n!} \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$,
$j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|, \end{equation} where
$\|\cdot\|$ represents the operator norm. This inequality is a special case of
a recent conjecture by Recht and R\'e.
