The linear upper-bound conjecture for almost-equidistant sets

Let $f_{\mathrm{ae}}(d)$ denote the maximum cardinality of an almost-equidistant set in $\mathbb R^d$, meaning a set in which among every three points, some pair is at distance $1$. Linear upper-bound conjecture. One has

$$f_{\mathrm{ae}}(d)=O(d).$$

The paper presents this as a natural conjecture that its methods do not prove; it predicts a linear, rather than superlinear, upper bound for the largest almost-equidistant sets.