The jump-separation conjecture for generalized directed hypergraph theories

Let $T'$ and $T$ be $r$-ary generalized directed hypergraph theories for $r \geq 3$. Write $J_T$ for the set of jumps of $T$ and $m_T$ for its multiplicity parameter. Jump-separation conjecture. If $J_{T'} \subseteq J_T$ and $m_T=2m_{T'}$, then there exists some $\alpha\in[0,1)$ that is a jump for $T$ but not for $T'$. The conjecture is false when $r=2$, since every $\alpha\in[0,1)$ is a jump for digraphs; its validity for $r\geq3$ remains open.