What is known about the question (Q) below?

Let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid, that is, $(M, \cdot)$ is the free monoid on an alphabet $X$ and $\le$ is a total order on $M$ such that $xy < xz$ and $yx < zx$ for all $x,y,z \in M$ with $y < z$.

(Q) Does there always exist an embedding of $\mathbb M$ into a linearly ordered group? In more plain words: do there always exist a linearly ordered group $\mathbb G = (G, \cdot, \le)$ and a (monoid) monomorphism $f: (M, \cdot) \to (G, \cdot)$ such that $f(x) < f(y)$ for all $x,y \in M$ with $x < y$?

The answer is affirmative in the case when $\le$ is the lexicographic order induced on $(M, \cdot)$ by any well-ordering of the alphabet $X$ (this can be proved, e.g., by the "Magnus trick").

But what about the rest? Any reference?