Conjecture on isometric transitivity of norming order units

Let $(V,V^+,e)$ be an order unit space. An element $u\in V$ with $\lVert u\rVert=1$ is called a norming order unit if it has the order-unit property and norms the space as specified in the source.

Isometric transitivity conjecture. If $u\in V$ with $\lVert u\rVert=1$ is a norming order unit, then there exists a surjective linear isometry

$$T:V\to V$$

such that $T(e)=u$.

This conjecture asks whether every norming order unit lies in the orbit of the distinguished order unit under the surjective linear isometry group of $V$. Its status is not resolved by the supplied text.