Lorentzian character conjecture for simple highest weight modules

Let $\mathfrak{g} = \oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$ be a direct sum of special linear Lie algebras, with positive roots $\Delta$ as above, and let $d = \sum_{t=1}^T (n_t+1)$. For an arbitrary weight $\lambda \in \mathfrak{h}^*$, let $M(\lambda) \twoheadrightarrow V(\lambda)$ be the unique simple highest weight module with highest weight $\lambda$. For $\delta \in \mathbb{N}^{d}$, let $N(x^\delta \cdot \operatorname{char} V(\lambda))$ denote the normalized shifted character. Lorentzian character conjecture. For every such $\lambda$ and every $\delta \in \mathbb{N}^{d}$, the normalized shifted character

$$N(x^\delta \cdot \operatorname{char} V(\lambda))$$

is Lorentzian. This extends the cited conjecture for simple highest weight modules to arbitrary, possibly non-integral, weights and to direct sums of special linear Lie algebras; the status of the assertion is not resolved in the supplied source context.