Dominance conjecture for nonzero Lie algebra word maps

Let $K$ be the base field, let $\mathfrak{g}$ be a semisimple $K$-Lie algebra, and let $w\in\mathcal{L}_{r}$ be a Lie algebra word. Let $\varphi_{w}:\mathfrak{g}^{r}\rightarrow\mathfrak{g}$ be the induced word map. Assume that $\varphi_{w}$ is non-zero, meaning

$$\varphi_{w}(\mathfrak{g}(\overline{K})^{r})\neq\{0\}.$$

Dominance conjecture. Then $\varphi_{w}$ is dominant. This extends the stated Lie-algebra analogue of Borel's theorem beyond the explicitly assumed $\mathfrak{sl}_{2}$ condition; the supplied text does not state whether the conjecture has been resolved.