Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $|.|$ the word length with respect to this set $S$. Is the function $k(g,h) = \frac{1}{1+|gh^{-1}|}$ positive definite, for $g,h \in G$?
Motivation: This matrix $k(g,h)$ plays a role in a group theoretic reformulation of the Lagarias inequality: https://mathoverflow.net/questions/330359/what-properties-characterize-the-function-lx-x-expx-logx
Also asked on MO, since it may be research relevant: https://mathoverflow.net/questions/391956/is-the-function-kg-h-frac11gh-1-positive-definite