The integral conformal circuit decomposition conjecture

Let $W a a$ be a rational linear subspace. For $z a a a W\cap\mathbb{Z}^n$, a conformal circuit decomposition is a representation of $z$ as a \sum of conformal circuit vectors. A vector is $1/\dot\kappa_W$-integral when its coordinates lie in $(1/\dot\kappa_W)\mathbb{Z}$. Integral conformal circuit decomposition conjecture. For every $z\in W\cap\mathbb{Z}^n$, there exists a conformal circuit decomposition

$$z=\sum_{k=1}^h g^k,$$

with $h\le n$, such that every $g^k$ is a $1/\dot\kappa_W$-integral vector in the set of elementary vectors $\mathcal F(W)$. The conjecture generalizes the stated integer decomposition property for the case $\kappa_W=\dot\kappa_W=1$; the source also notes an equivalent formulation for vectors in the Graver basis, but gives no resolution status.