Diehl–Reizenstein conjecture on polynomial invariants of path orbits

Let $G$ be a subgroup of $ame{SL}_d(\mathbb{R})$, and let $Z,Z':[0,T]\to\mathbb{R}^d$ be curves. Suppose that

$$\langle \operatorname{IIS}(Z),\varphi\rangle=\langle \operatorname{ISS}(Z'),\varphi\rangle$$

for every $\varphi\in T(\mathbb{R}^d)$ fixed by $\tilde{\phi}_{A^\top}$ for every $A\in G$. Diehl–Reizenstein conjecture. There exist $A\in G$ and a curve $\bar Z$ tree-like equivalent to $Z$ such that

$$A\bar Z=Z'.$$

The conjecture asserts that polynomial invariants separate $G$-orbits of paths up to tree-like equivalence. Its general status is not specified in the source.