Echelonmotion realization of Dilworth's theorem for modular lattices

Let $L$ be a modular lattice, and let $\sigma$ be a linear extension of $L$. For $x\in L$, write $\operatorname{Cov}_L^{\uparrow}(x)$ for the elements covering $x$ and $\operatorname{Cov}_L^{\downarrow}(x)$ for the elements covered by $x$; let $\operatorname{Ech}_\sigma$ denote echelonmotion with respect to $\sigma$. Echelonmotion realization conjecture. For every $x\in L$, we have

$$\left|\operatorname{Cov}_L^{\uparrow}(\operatorname{Ech}_\sigma(x))\right|=\left|\operatorname{Cov}_L^{\downarrow}(x)\right|.$$

This conjecture asks for echelonmotion to give an explicit bijective realization of Dilworth's theorem for modular lattices, which equates the numbers of elements covering $k$ elements and covered by $k$ elements. Its resolution is not indicated in the supplied text.