Birman's conjecture for singular braids on closed surfaces

Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be the braid group of $M$ on $n$ strings, and let $S{B}_n(M)$ be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map $\eta :S{B}_n(M)\to \mathbb{Z}[B_n(M)]$, introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.