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 $r\le n/2$, we prove $$T(n,r)=O\!\left(\frac{r2^n}{n+1}\right).$$ We also obtain lower bounds in various regimes of $r$ as a function of $n$.