The period-three coexistence conjecture for special alpha-limit sets

Let $f:[0,1]\to[0,1]$ be continuous, let $x\in[0,1]$, and let $s\alpha(x)$ be the special $\alpha$-limit set of $x$. Period-three coexistence conjecture. If $s\alpha(x)$ contains a periodic orbit of period $3$, then, for every positive integer $n$, $s\alpha(x)$ contains a periodic orbit of period $n$ or $2n$. The paper presents this as an unresolved conjecture about the coexistence of periodic orbits inside special $\alpha$-limit sets.