Why I can numerically use signatures on rough paths?

Question

In the paper deep signature transforms ({bonnier, kidger, perez, salvi, tlyons}) they use the signatures on Brownian motion and they invert it with the inversion method defined in (The insertion method to invert the signature of a path Adeline Fermanian, Jiawei Chang, Terry Lyons and Gerard Biau). The issue for me is that the inversion (or even the signature) is actually well defined for bounded variation paths where as Brownian motion does not have bounded variation. I cannot figure out why we can apply this method to Brownian motion. What I thought was that we can apply this method numerically as numerically generated Brownian motion has bounded variation as it is essentially linearly interpolated increments of standard random variable. But I am not sure if this is correct. thanks a lot for some clarification.