Optimal validity region for the Lorentzian polynomial criterion

Determine the exact set of positive integer pairs (a,b) for which the Hessian of \(\partial_x f\), where \(f(x_1,x_2,x,y)=(ax+y)[(bx+x_1+y)(bx+x_2+y)+xy]\), is Lorentzian, and identify conditions under which \(f\) itself is Lorentzian beyond the sufficient cases \(a=1\) and \(a=b\).

Background

Lemma 3.2 proves that the polynomial $f(x_1,x_2,x,y)=(ax+y)[(bx+x_1+y)(bx+x_2+y)+xy]$ is Lorentzian when $a=1$ or $a=b$.

The authors explicitly state that these hypotheses are sufficient but not necessary. They report numerical evidence that the Hessian is Lorentzian for $(a,b)=(3,2)$ and $(4,2)$, but not for $(2,1)$ or $(5,2)$, and leave the exact region of validity unresolved.