Rubin's conjecture on saturated transfer systems for cyclic groups

Fix positive integers $r_1,\ldots,r_k$, and let $p_1,\ldots,p_k$ be distinct primes. A saturated transfer system on a finite group is a transfer system satisfying the two-out-of-three property: whenever $L\leq K\leq H\leq G$ and two of $L\to K$, $L\to H$, and $K\to H$ hold, so does the third. Rubin's conjecture. For sufficiently large distinct primes $p_1,\ldots,p_k$, every saturated transfer system on

$$C_{p_1^{r_1}\cdots p_k^{r_k}}$$

can be realized by a linear isometries operad. Linear isometric transfer systems are known to form a subclass of the saturated transfer systems, and examples show that equality does not hold in general; Rubin's conjecture predicts equality for cyclic groups with sufficiently large prime divisors.