Kirillov's non-negativity conjecture for Hecke–Grothendieck and generalized key polynomials

Let $n$ be a positive integer, let $w\in S_n$ be a permutation, let $\zeta$ be a weak composition, and let $\boldsymbol{x}=(x_1,\ldots,x_n)$. Write $\mathcal{KN}_w^{(\beta,\alpha,\gamma)}(\boldsymbol{x};\boldsymbol{0})$ for the generalized Hecke–Grothendieck polynomial and $K_\zeta^{(\beta,\alpha,\gamma)}(\boldsymbol{x})$ for the generalized key polynomial. Kirillov's conjecture. The polynomials have non-negative coefficients:

$$\mathcal{KN}_w^{(\beta, \alpha, \gamma)}(\boldsymbol{x}; \boldsymbol{0}) \in \mathbb{N}[\beta, \alpha, \gamma][x_1, x_2, \ldots, x_n], \qquad K_\zeta^{(\beta, \alpha, \gamma)}(\boldsymbol{x}) \in \mathbb{N}[\beta, \alpha, \gamma][x_1, x_2, \ldots, x_n].$$

The conjecture is disproved as stated, although the associated lattice model gives insight into the strongest form that does hold; non-negativity is established for various specializations, including $\gamma=0$.