Next stage: bounded ratios and optimal bounding constants for ternary Lorentzian polynomials

A possible next step would be to read some of the more recent work on volume polynomials (June Huh has a survey paper).

For someone who would like to focus on the polynomial theory perspective, Lorenztian polynomial is closely related to stable polynomials, which is another active research topic.

As to some concrete problems, one can check Daoji Huang, June Huh, Daniel Soskin, Botong Wang, Bounded ratios for Lorentzian matrices.

There are some conjectures there, and it's encouraged to try to study similar questions for degree 3 Lorentzian polynomial in 3 variables. The latter is a hands-on problem, but will not be very conceptually interesting.