The extremal bound for almost-equidistant diameter sets

Let an almost-equidistant diameter set be a set of points in $\mathbb R^d$ of diameter $1$ in which every three points contain a pair at distance $1$. The extremal bound conjecture. An almost-equidistant diameter set in $\mathbb R^d$ has at most

$$\left\lfloor\frac{3(d+1)}{2}\right\rfloor$$

points. The conjecture is motivated by a construction from two disjoint cliques in a diameter graph, with $d+1$ and $\left\lfloor\frac{d+1}{2}\right\rfloor$ vertices; its resolution status is not established by the supplied evidence.