The Lorentzian conjecture for normalized shifted characters

Let $\Lambda$ be the weight lattice of $\mathfrak{sl}_m(\mathbb{C})$, let $\mathrm{V}(\lambda)$ be the irreducible module of highest weight $\lambda$, and define its shifted character by

$$\mathrm{ch}_{\lambda}(x_1,\ldots,x_m)=\sum_{\mu\in\Lambda}\dim\mathrm{V}(\lambda)_\mu\,x^{\mu-\lambda}.$$

For $\delta\in\mathbb{N}^m$, write $x^\delta=x_1^{\delta_1}\cdots x_m^{\delta_m}$. Shifted-character conjecture. The polynomial

$$\mathrm{N}(x^\delta\,\mathrm{ch}_{\lambda}(x_1,\ldots,x_m))$$

is Lorentzian for every $\lambda\in\Lambda$ and $\delta\in\mathbb{N}^m$. The source reports computational tests for selected weights and dimensions, but no proof of the general assertion.