Non-subloop conjecture for Moufang elements in PACC-loops

Let $Q$ be a power-associative conjugacy-closed loop (PACC-loop), and let $M(Q)$ denote the set of Moufang elements of $Q$.

Non-subloop conjecture. There exists a PACC-loop $Q$ in which $M(Q)$ is not a subloop.

The claim asks whether the Moufang elements of every PACC-loop necessarily form a subloop; the authors state that they have not been able to prove this even in the power-associative case and therefore propose the existence assertion as an open conjecture.