Unitary representability conjecture for free topological groups

Let $X$ be a Tikhonov space, let $F(X)$ denote the free topological group on $X$, and let $M$ be a metric space of $L^1$-type, meaning that $M$ is isometric to a subspace of $L^1_{\mathbf{R}}(\mu)$ for some measure space $(\Omega,\mu)$.

Unitary representability conjecture. For every Tikhonov space $X$, there exists a metric space $M$ of $L^1$-type such that $F(X)$ is isomorphic to a subgroup of the group of isometries $\operatorname{Iso}(M)$.

This is proposed as a non-abelian analogue of the preceding unitary-representability results for additive groups of $L^1$ spaces. Its status is not determined by the supplied text.