Bernoulli-moment characterization conjecture for distinguished matrices

Fix $n\in\mathbb{N}$. Let $T_n^{S^1}(\mathbb{R})$ be the set of upper triangular real matrices with diagonal entries $1$ for which the eigenvalues of $\,S^{-1}S^t$ lie on the unit circle, and let $T_n^{uni}(\mathbb{Z})$ be the corresponding integral unipotent triangular matrices. For a matrix $S\in T_n^{S^1}(\mathbb{R})\cap T_n^{uni}(\mathbb{Z})$, suppose the Cecotti–Vafa recipe supplies an ordered spectrum $\operatorname{Sp}(S)=(\alpha_1,\ldots,\alpha_n)$, and let $\Gamma^{Ber}_{2k}$ denote its Bernoulli moments.

Bernoulli-moment characterization conjecture. There is a family of functions $r(k,n,\nu):\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ such that a matrix $S$ is a distinguished matrix of a singularity if and only if its Coxeter–Dynkin diagram $CDD(S)$ is connected and, for every $k\in\mathbb{N}$,

$$0\leq(-1)^k\Gamma^{Ber}_{2k}(\operatorname{Sp}(S),\alpha_n-\alpha_1)\leq r(k,n,\alpha_n-\alpha_1).$$

The conjecture is intended to characterize distinguished matrices through the full sequence of Bernoulli-moment inequalities. The paper presents it as part of a tower of conjectures and does not establish the required bounds or the converse characterization in the indefinite cases.