The Lorentzian conjecture for normalized key polynomials

Let $\alpha\in\mathbb{N}^n$ be a composition, and let $\kappa_\alpha(x_1,\ldots,x_n)$ be its key polynomial. Key-polynomial conjecture. The normalized polynomial

$$\mathrm{N}(\kappa_\alpha(x_1,\ldots,x_n))$$

is Lorentzian for every composition $\alpha$. The source reports tests for small compositions; the Lorentzian assertion remains open, although M-convexity of the support is known.