Existence conjecture for Johnson statistics

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let $\mathcal{R}$ be its root sublattice, and let $\mathsf{T}(x_1,\ldots,x_{a-1})=x_1+2x_2+\cdots+(a-1)x_{a-1}$. A Johnson statistic is a function $\mathsf{J}:\mathcal{R}\to\mathbb{Z}$ such that

$$\mathsf{J}(\mathbf{x}+a\mathbf{y})=\mathsf{J}(\mathbf{x})+a\mathsf{T}(\mathbf{y})$$

for $\mathbf{x}\in\mathcal{R}$ and $\mathbf{y}\in\mathcal{L}$, and

$$\operatorname{Cat}(a,b)_q=\sum_{\mathbf{x}\in\mathcal{R}\cap b\Delta}q^{\mathsf{J}(\mathbf{x})}$$

for every $b\geq0$ coprime to $a$. Johnson statistic existence conjecture. There exists at least one Johnson statistic for each value of $a$. Such a statistic would provide a direct lattice-point generating-function interpretation of rational $q$-Catalan numbers; the source presents the conjecture as open and gives indirect evidence.