The Lorentzian conjecture for normalized homogeneous Grothendieck polynomials

Let $w\in\mathcal{S}_n$ be a permutation, let $\ell(w)$ and $d(w)$ denote the degrees specified in the source, and define the homogeneous Grothendieck polynomial

$$\widetilde{\mathfrak{G}}_w(x_1,\ldots,x_n,z)=\sum_{k=0}^{d(w)-\ell(w)}(-1)^k\mathfrak{G}^k_w(x_1,\ldots,x_n)z^{d(w)-\ell(w)-k}.$$

Homogeneous-Grothendieck conjecture. The normalized polynomial

$$\mathrm{N}(\widetilde{\mathfrak{G}}_w(x_1,\ldots,x_n,z))$$

is Lorentzian for every $w\in\mathcal{S}_n$. The source reports tests for small symmetric groups and notes that this conjecture would imply the homogeneous-component conjecture; the general assertion remains open.