Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators.

Preliminaries: Recall that we say that $s, t \in S$ are $\mathcal{L}$-related if $s^{-1}s = t^{-1}t$. Given an $\mathcal{L}$-class $L \subset S$, we may construct its' Schützenberger graph $\Lambda(L, K)$, whose vertices are the points of $L$ and where $x, y \in L$ are connected by an edge labeled by $k \in K$ if $kx = y$. We consider $L$ equipped with the natural path metric $d_L$ via $\Lambda(L, K)$. Another studied congruence in $S$ is $\sigma$, where $s \sigma t$ if $sx = tx$ for some $x \in S$. The quotient $S/\sigma$ is a group $G$ known as the maximal homomorphic image of $S$. Moreover, we say that $S$ is E-unitary if whenever $s \sigma t$ and $s \mathcal{L} t$ then $s = t$, i.e. the quotient map embeds every $\mathcal{L}$-class into $G$.

Question: Let $S = \langle K \rangle$ be a fin. gen. E-unitary inverse semigroup. Let $L \subset S$ be an $\mathcal{L}$-class. Is the quotient map $L \rightarrow G$ a quasi-isometric embedding? That is, are there constants M, C > 0 such that for all $x, y \in L$ $$ \frac{1}{M} d_L(x, y) - C \leq d_G(x\sigma, y\sigma) \leq M d_L(x, y) + C$$ where $d_G$ is the path metric in the left Cayley graph of $G$ with respect to the generating set $K \sigma$. Observe that the right inequality above is true for any $M \geq 1$, since any geodesic between $x, y$ falls down to a path between $x\sigma, y\sigma$.

Partial results/remark: it's clear that if $L$ has only finitely-many $\mathcal{R}$-classes then the quotient map is going to be a quasi-isometry. Indeed, the $\mathcal{H}$-class of the idempotent of $L$ is a group included in $G$, and that inclusion of groups is a quasi-isometry. Since $L$ has only finitely-many $\mathcal{R}$-classes, then so is the inclusion of $L$ into $G$, i.e., the quotient map.

Motivation: In [1] quasi-isometries in monoids are studied, from the point of view of the Cayley graph. However, little is said about my inquiry, and I haven't been able to find any reference on this in the literature. My guess is the answer should be yes, but any help is greatly appreciated.

[1] Gray and Kambites, Groups acting on semimetric spaces and quasi-isometries of monoids, Trans. Ame. Math. Soc. 365 (2013) 555--578.