The balanced orbit's product-majorization conjecture

Let $1\leq p<q$ be coprime integers. For a word $w\in\mathbb{W}_{p,q}$, let $\mathcal{I}(w)=(w_1,\ldots,w_q)$ be its base-2 orbit and define the partial products by

$$\mathcal{P}_i(w)=\prod_{k=1}^{i}\mathcal{I}_k(w).$$

The relation $w'\prec_p w$ means that $\mathcal{P}_i(w')\geq \mathcal{P}_i(w)$ for every $1\leq i\leq q$. Let $b\in\mathbb{W}_{p,q}$ be the unique balanced orbit. The balanced orbit's product-majorization conjecture. The orbit $b$ is the least element of $(\mathbb{W}_{p,q},\prec_p)$; equivalently, for every $w\in\mathbb{W}_{p,q}$,

$$\mathcal{P}_i(b)\geq\mathcal{P}_i(w)\quad\text{for all }1\leq i\leq q.$$

This is the product-majorization analogue of Jenkinson's theorem that the balanced orbit is least under partial-sum majorization. The paper says that it is very likely to be true and suggests that a proof might use similar ideas, but it remains unproved here.