Realizability conjecture for compatible pairs of transfer systems

Let $G$ be a finite group. A pair of $G$-transfer systems is called compatible when it satisfies the compatibility conditions for a pairing of $N_{\infty}$-operads, and it is realizable if there exist $N_{\infty}$-operads whose associated transfer systems are the given pair and which admit a pairing. Realizability conjecture. Every compatible pair of $G$-transfer systems is realizable. This conjecture asks for a converse to the homotopical obstruction to pairings of $N_{\infty}$-operads. Although every individual transfer system is realized by an $N_{\infty}$-operad, realizing a compatible pair is subtler because pairings of operads are not homotopy-invariant; the conjecture remains open in the supplied text.