Nonemptiness conjecture for arithmetic triangle triples

For each integer $r\geq 1$, let $\mathcal{T}(r)$ denote the set of $r$-arithmetic triples. Nonemptiness conjecture. The set $\mathcal{T}(r)$ is nonempty for all $r\geq 1$. The preceding computations give nonempty sets for every $r\leq 15$, but no general proof or convincing growth rate for $\#\mathcal{T}(r)$ is provided.