Erdős Problem #982 — Distinct distances from a vertex of a convex polygon

Let $n\ge3$ distinct points $p_0,\ldots,p_{n-1}\in\mathbb R^2$ form a convex polygon. Must there exist a vertex $p_i$ having at least $\lfloor n/2\rfloor$ distinct distances from the other vertices; that is, must there exist $i$ such that

$$\#\{d\in\mathbb R:\exists j\ne i\text{ with }d=\operatorname{dist}(p_i,p_j)\}\ge\left\lfloor\frac n2\right\rfloor?$$