The converse characterization of central weightings

Let ${\mathcal{S}}$ be a step set, let ${\mathfrak a}$ be a positive weighting, and let $Q_{\mathfrak a}(x_1,\dots,x_d;t)$ and $Q(x_1,\dots,x_d;t)$ denote the corresponding weighted and unweighted generating functions. A weighting is central when there are constants $\alpha_1,\dots,\alpha_d,\beta$ such that

$$a_s=\beta\prod_{k=1}^d\alpha_k^{\pi_k(s)}.$$

The converse characterization of central weightings. If there exist constants $\alpha_1,\dots,\alpha_d,\beta$ such that

$$Q_{\mathfrak a}(x_1,\dots,x_d;t)=Q(\alpha_1x_1,\dots,\alpha_dx_d;\beta t),$$

then the weighting ${\mathfrak a}$ is central and

$$a_s=\beta\prod_{k=1}^d\alpha_k^{\pi_k(s)}.$$

This would imply uniqueness of the function satisfying the kernel equation for a given model; in dimension two, it would imply uniqueness of the kernel solution. The claim is stated as open here, although it is known for random walks arising from root systems of Lie algebras.