Modified Lyons–Sidorova conjecture on entire log-signatures of tree-reduced paths

Let $V$ be a finite-dimensional real normed vector space, and let $\gamma$ be a continuous tree-reduced bounded-variation path in $V$. Write $S(\gamma)$ for its signature and $R(\log S(\gamma))$ for the radius of convergence of its logarithmic signature. A path is path-conjugate to a line segment if, after translation to its initial point, it is equivalent under reduced path concatenation to a path of the form $\alpha*\lambda_v*\overleftarrow{\alpha}$, where $v$ is its endpoint displacement and $\alpha$ is a based bounded-variation path.

Modified Lyons–Sidorova conjecture. One has

$$R\bigl(\log S(\gamma)\bigr)=\infty \quad\Longleftrightarrow\quad \gamma\text{ is path-conjugate to a line segment.}$$

The naive assertion that infinite radius characterizes literal straightness fails because infinite radius is invariant under conjugation. For tree-reduced paths, the source indicates that path conjugacy and signature conjugacy agree; the conjecture identifies infinite logarithmic-signature radius with conjugacy to a line, but its general resolution is not stated.