Characteristic-independence conjecture for level h-vectors

Let $h$ be an $h$-vector. We say that $h$ is level in characteristic $p$ (where $p\geq 0$) if there exists a level algebra $k[x_1,\dots,x_r]/I$ with $h$-vector $h$ and $\operatorname{char}(k)=p$.

Characteristic-independence conjecture. An $h$-vector is level in some characteristic if and only if it is level in every characteristic.

This conjecture asserts that the existence of a level algebra with a given $h$-vector is independent of the characteristic of the base field. The paper notes that all of its results are characteristic-free, but does not establish this more general statement.