The configuration space $C_n(M)$ of $n$ particles in some connected graph $M$ (thought of as the topological realisation of a one-dimensional CW-complex) is $$M^n \backslash \{ (x_1, \ldots, x_n) \mid x_i=x_j \ \text{for some} \ i \neq j\},$$ and the corresponding unordered configuration space $UC_n(M)$ is the quotient $C_n(M)/ \mathfrak{S}_n$ where $\mathfrak{S}_n$ acts freely on $C_n(M)$ by permuting the coordinates. The braid group $B_n(M)$ is the fundamental group of $UC_n(M)$ (which is connected if $M$ is connected itself).

Is it true that $B_{m}(M)$ admits an injective homomorphism into $B_n(M)$ if $m \leq n$? Is it true at least if $m \leq 2$?

It is worth noticing that the question is not trivial if we replace the graph $M$ with a sphere according to this question.