The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths

For a continuous bounded-variation path with signature $g$, logarithmic signature $l$ and increment $v$, the modified Lyons–Sidorova conjecture predicts the structure of $g$ when $R(l)=\infty$. The paper proves it: $g=1$ when $v=0$, and otherwise a prefix $\alpha$ of the centred path gives $S(\gamma) = S(\alpha)e^{v}S(\alpha)^{-1}$.