Rubin's saturation conjecture for linear isometries operads

Let $G$ be the finite Abelian group

$$G=G_{p_1^{m_1}\cdots p_n^{m_n}},$$

where $p_1,\ldots,p_n$ are distinct primes and $m_1,\ldots,m_n\in\mathbb{N}$. A saturated transfer system on $G$ is a transfer system with the property that whenever $(K,H)$ belongs to it and $K\leqslant L\leqslant H$, then $(L,H)$ also belongs to it. A transfer system is realized by a linear isometries operad if it is the transfer system associated to such an operad.

Rubin's saturation conjecture. For every $n$ there exists some $f(n)$ such that, for any collection of distinct primes $p_1,\ldots,p_n\geqslant f(n)$ and any choice of $m_1,\ldots,m_n\in\mathbb{N}$, every saturated transfer system on $G=G_{p_1^{m_1}\cdots p_n^{m_n}}$ is realized by a linear isometries operad.

This conjecture concerns when the special class of transfer systems arising from linear isometries operads accounts for all saturated transfer systems. The statement is known for cyclic $p$-groups and, for groups of order $p^mq^r$ with distinct primes, the preceding results establish the relevant two-prime case when the primes are sufficiently large; the general assertion for arbitrary $n$ remains open.