Asymptotic persistence of the experimentally observed improvement

Establish whether the greater improvement suggested by numerical experiments for the multivariate spectrahedral relaxation persists for higher indices n, and determine which phenomena, if any, could prevent its generalization.

Background

The earlier linearization using the vector sequence {(y,1,−1)}\{(y,1,\mathbf{-1})\} certifies only a small improvement, whereas numerical experiments for small values of nn suggest that the multivariate relaxation may provide a substantially larger gain.

The authors explicitly note that computations become difficult for large indices and leave unresolved whether the observed improvement continues asymptotically or is obstructed by a phenomenon that appears only at higher dimensions.

References

However, it is not clear what phenomena could actually hinder the generalization of that greater improvement for higher indices $n$.

— Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets  (2507.18434 - Nevado, 24 Jul 2025) in Remark 3, “Plenty of room for improvement,” item 1, Section 2, Overview