Lorentzianity conjecture for normalized dual $k$-Schur polynomials

Let $f=\sum_{\alpha\in\mathbb{N}^n}c_{\alpha}x^{\alpha}$ be a polynomial, and define its normalization by

$$N(f)=\sum_{\alpha\in\mathbb{N}^n}c_{\alpha}\frac{x_1^{\alpha_1}}{\alpha_1!}\cdots\frac{x_n^{\alpha_n}}{\alpha_n!}.$$

For $\lambda\in \mathrm{Par}^k(d)$, let $\mathfrak{S}_{\lambda}^{(k)}$ denote the corresponding dual $k$-Schur polynomial. Lorentzianity conjecture. For any $\lambda\in \mathrm{Par}^k(d)$, the polynomial $N(\mathfrak{S}_{\lambda}^{(k)})$ is Lorentzian. This is proposed as the dual $k$-Schur analogue of the known Lorentzianity of normalized Schur polynomials; no resolution is given here.