2 problems

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Weak real zero amalgamation conjecture for polynomials on two-element overlaps
Weak real zero amalgamation conjecture. There should be a real zero polynomial $Q\in\mathbb{R}[x_i\mid i\in E']$ such that, for $k=1,2$ ,
Conjecturesolvedmatroids polynomial-amalgamation real-zero-polynomialsarXiv posted 2 days ago
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Schweighofer–Sawall weak real zero amalgamation conjecture
Schweighofer–Sawall's weak real zero amalgamation conjecture. There should be a real zero polynomial $H\in\mathbb{R}[x_1,x_2,y_1,\ldots,y_m,z_1,\ldots,z_n]$ such that
Conjectureopenmatroids polynomial-amalgamation real-zero-polynomialsarXiv posted 2 days ago

Weak real zero amalgamation conjecture for polynomials on two-element overlaps