Conjecture on spectral numbers and spectral pairs for triangular matrices

Let $T(n,\mathbb R)$ be the space of upper triangular matrices under consideration. A Seifert form stratum is a union of components of a fiber of the map to isomorphism classes of Seifert form pairs, and an eigenvalue stratum is a union of components of a fiber of the map to unordered monodromy eigenvalues, with components permuted transitively by the sign group. For $S\in T(n,\mathbb R)$, write $\operatorname{Sp}(S)$ for its spectral numbers and $\operatorname{Spp}(S)$ for its spectral pairs. Spectral-strata conjecture. (a) $T_{{\rm HOR}1}(n,\mathbb R)$ intersects each eigenvalue stratum in $T(n,\mathbb R)$. (b) If $S_1,S_2\in\bigcup_{k=1,2}T_{{\rm HOR}k}(n,\mathbb R)$ lie in the same eigenvalue stratum, then $\operatorname{Sp}(S_1)=\operatorname{Sp}(S_2)$. (c) If $S_1,S_2\in\bigcup_{k=1,2}T_{{\rm HOR}k}(n,\mathbb R)$ lie in the same Seifert form stratum, then $\operatorname{Spp}(S_1)=\operatorname{Spp}(S_2)$. If true, these assertions would extend spectral numbers and spectral pairs from HOR-matrices to the relevant strata of the full triangular-matrix space; the source gives no resolution.