The monoid realization conjecture for pre-upho colored lattices

Let $L$ be a finite graded lattice whose maximum element is the join of its atoms, and let $c$ be a pre-upho coloring of $L$. Let $s_1,\ldots,s_r$ be the atoms of $L$, and define the monoid $M$ by the generators $s_1,\ldots,s_r$ and the relations equating the color words along every pair of saturated chains from $\hat{0}$ to the join $x_1\vee y_1$ of two atoms $x_1,y_1$:

$$M = \left\langle s_1,\ldots,s_r\mid c(\hat{0}=x_0\lessdot x_1)c(x_1\lessdot x_2)\cdots c(x_{k-1}\lessdot x_k = x_1 \vee y_1) = c(\hat{0}=y_0\lessdot y_1)c(y_1\lessdot y_2)\cdots c(y_{k-1}\lessdot y_k = x_1 \vee y_1) \right\rangle.$$

Monoid realization conjecture. The monoid $M$ is left-cancellative, every pair of elements of $M$ has a greatest common left divisor, and therefore $\mathcal{L} \coloneqq (M,\leq_L)$ is an upho meet semilattice containing a rank-preserving embedded copy of $L$. This would establish the converse direction: every pre-upho coloring gives rise to a colored upho lattice having $L$ as its core. The source presents this as a speculation; no resolution is supplied.