For even n, a carefully chosen sequence of vectors makes the spectrahedral relaxation bound for Eulerian polynomial roots exceed the univariate bound by asymptotically (3/8)(9/8)^{n/2}.
Wag- ner, Proof of the monotone column permanent conjecture , in *Notions of Positivity and the Geometry of Polynomials*, Pages 63–78, Springer, 2011
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Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets
For even n, a carefully chosen sequence of vectors makes the spectrahedral relaxation bound for Eulerian polynomial roots exceed the univariate bound by asymptotically (3/8)(9/8)^{n/2}.