The thermomajorization characterization by fixed-point stochastic maps

Let $x,y,d\in\mathbb R^n$ with $d\geq 0$ be given. A matrix $A\in\mathbb R_+^{n\times n}$ is column-stochastic when $\mathbbm e^\top A=\mathbbm e^\top$; it fixes $d$ when $Ad=d$. Thermomajorization characterization. The following are equivalent: there exists a column-stochastic matrix $A$ fixing $d$ and satisfying $Ay=x$; and

$$\mathbbm e^\top x=\mathbbm e^\top y$$

and

$$\lVert d_i x-y_i d\rVert_1\leq\lVert d_i y-y_i d\rVert_1 \qquad\text{for all }i=1,\ldots,n.$$

This would characterize the transformation relation $x\prec_d y$ by finitely many norm inequalities. The paper states that the result proved there is only a special case, that numerous examples suggest the conjecture, and that it is not attempted because it is beside the article's main point; its resolution is therefore left open.