Triangle-free subsets of the $r$-distance graph of the Hypercube

Given the $r$-distance graph on the hypercube ${\mathbb{F}}_2^n$, where two vertices are adjacent if their Hamming distance is exactly $r$, we study the maximum size $T(n,r)$ of a triangle-free set of vertices. For even $r\le n\mathrm{/}2$, we prove $$T(n,r)=O\text{\hspace{0.17em}⁣}\left(\frac{r{2}^n}{n+1}\right)\mathrm{.}$$ We also obtain lower bounds in various regimes of $r$ as a function of $n$.