Fajtlowicz's conjecture on average neighbour degree and distance spectra

Let $G$ be a finite simple connected graph of order $n=|V(G)|\ge2$. For each vertex $v$, let $d(v)$ be its degree and $N(v)$ its open neighbourhood, and define

$$d^*(v)=\frac1{d(v)}\sum_{u\in N(v)}d(u),\qquad \delta^*(G)=\min_{v\in V(G)}d^*(v).$$

Let $D(G)$ be the distance matrix of $G$, and write $\lambda_{\min}(D(G))$ for its smallest eigenvalue. Fajtlowicz's conjecture. If $G$ has order at least three and girth at least five, then

$$\delta^*(G)\le-\lambda_{\min}(D(G)).$$

This is a Graffiti conjecture attributed to Fajtlowicz's 1998 Written on the Wall report and recorded by Aouchiche and Hansen. Its resolution is not established by the supplied text.