A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces

A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $\sigma (T)=\{1\}$, the inequality $T\ge \mathrm{id}$ holds true. While the problem remains unsolved in its entirety, a positive solution is known in finite dimensions. In the broader context of ordered Banach spaces, Drnovšek provided an infinite-dimensional counterexample. In this note, we demonstrate the existence of finite-dimensional counterexamples, specifically on the ice cream cone and on a polyhedral cone in ${\mathbb{R}}^3$. On the other hand, taking inspiration from the notion of $m$-isometries, we establish that each counterexample must contain a Jordan block of size at least $3$.