Conjecture on lifting spectral numbers to spectral pairs

Let $S$ be a matrix in a Seifert form stratum of $T(n,\mathbb R)$ that is not met by $\bigcup_{k=1,2}T_{{\rm HOR}k}(n,\mathbb R)$. Let $\operatorname{Sp}(S)$ and $\operatorname{Spp}(S)$ denote its spectral numbers and spectral pairs. Lifting conjecture. The spectral numbers $\operatorname{Sp}(S)$ lift in a natural way to the spectral pairs $\operatorname{Spp}(S)$. The claim would provide spectral-pair data for Seifert form strata not reached by the HOR-matrix families. The source gives no resolution.