The maximal-path ideal conjecture for staged trees

Let $\mathcal{T}$ be a staged tree, with parameterisation $\varphi$ and maximal-path ideal $I_{\mathrm{mpaths}}$, the ideal generated by path differences associated to maximal paths.

Maximal-path ideal conjecture. The kernel of $\varphi$ is exactly the ideal generated by maximal paths,

$$\ker \varphi=I_{\mathrm{mpaths}}.$$

The conjecture is motivated by the examples in the paper, all of which satisfy this equality. It would identify the distinguished prime component of the ideal of model invariants with the maximal-path ideal for every staged tree; the paper does not establish the claim in general.