Dense periodic points imply periodicity for monotone maps with polyhedral order cones

Let $K\subset\mathbb{R}^n$ be a solid, pointed, polyhedral closed convex cone, and let $X\subset\mathbb{R}^n$ be a nonempty set whose interior is connected and dense in $X$. Equip $\mathbb{R}^n$ with the partial order $y\succeq x$ if and only if $y-x\in K$. Let $T\colon X\to X$ be an injective continuous map that is monotone for this order, meaning that $x\succeq y$ implies $Tx\succeq Ty$. Write $\mathcal{P}_k(T)$ for the points of period $k$ and $\mathcal{P}(T)=\bigcup_k\mathcal{P}_k(T)$ for the set of periodic points.

Dense periodicity conjecture. If $\mathcal{P}(T)$ is dense in $X$, then $T$ is periodic: there is a positive integer $k$ such that $X=\mathcal{P}_k(T)$.

The conjecture asserts that dense periodic points force a single common period for this class of monotone dynamical systems. In the supplied text, the statement is presented as a speculation; no resolution is given there.