Cecotti–Vafa spectrum conjecture for distinguished matrices

Fix $n\in\mathbb{Z}_{\geq 2}$. Let $T_n^{S^1}(\mathbb{R})$ consist of upper triangular real matrices with diagonal entries $1$ whose matrices $S^{-1}S^t$ have all eigenvalues on the unit circle. For $S\in T_n^{S^1}(\mathbb{R})$, a spectrum is an ordered tuple $\operatorname{Sp}(S)=(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}^n$.

Cecotti–Vafa spectrum conjecture. The Cecotti–Vafa construction can be made precise so that

$$\alpha_1\leq\cdots\leq\alpha_n,\qquad \alpha_i+\alpha_{n+1-i}=0,$$

and $e^{-2\pi i\alpha_1},\ldots,e^{-2\pi i\alpha_n}$ are the eigenvalues of $S^{-1}S^t$.

The original construction requires natural choices of a path from the identity to $S$ and continuous lifts of the eigenvalue arguments. The paper explains that these choices present obstacles and that the construction is currently available only for certain subsets of the relevant matrix space.