Auchiche–Hansen's proximity and distance eigenvalue conjecture

Let $G$ be a connected graph on $n\geq 4$ vertices with diameter $D$, proximity $\pi$ and distance spectrum $\partial_1\geq \ldots \geq \partial_n$. Auchiche–Hansen's conjecture.

$$\pi + \partial_{\mathopen{}\mathclose\bgroup\originalleft\lfloor \frac{2D}{3} \aftergroup\egroup\originalright\rfloor} > 0.$$

This conjecture would strengthen a result of Merris on proximity and distance eigenvalues. It is refuted by a connected graph obtained from a path on 13 vertices by attaching sufficiently many pendant vertices; specifically, the construction gives a counterexample for all $n\geq 190$ pendant vertices.