The converse chordality conjecture for positive definite function extensions

Let $G$ be a group, let $S\subset G$, and let $\Gamma(G,S)$ be the associated graph. A function $\phi:S\to {\mathcal L}({\mathcal H})$ is positive definite if its corresponding kernels satisfy the positive-definiteness condition used for operator-valued functions.

Converse chordality conjecture. For every $S\subset G$ such that $\Gamma(G,S)$ is not chordal, there exists a positive definite function

$$\phi:S\to {\mathcal L}({\mathcal H})$$

which does admit a positive definite extension to $G$.

This conjecture proposes the reciprocal of the paper's extension theorem: nonchordality should guarantee the existence of a positive definite function on $S$ that extends positively to the whole group. The supplied text gives no evidence that the conjecture has been resolved.