The four-cycle conjecture for the abelian loop

Let $S$ be the abelian loop of odd numbers used in the paper, with binary operation $\bullet$. Consider quartets $(a,b,c,d)$ of distinct numbers in $S$ satisfying

$$a \bullet b=b \bullet c=c \bullet d=d \bullet a.$$

Four-cycle conjecture. There are infinitely many such quartets.

This conjecture concerns four-cycles in the operation table of the loop. The supplied text does not provide evidence resolving it, so its status remains open.