Steinberg-map compatibility conjecture for symmetric-pair orbit embeddings

Assume an orbit embedding

$$\iota:\mathfrak{X}/K\hookrightarrow\mathbb{X}/\mathbb{K}$$

and Steinberg maps

$$\Phi^{\pm\theta}:\mathfrak{X}/K\to\mathcal{N}_{\mathfrak{g}}^{\pm\theta}/K, \qquad \mathbbl{\Phi}^{\pm\theta}:\mathbb{X}/\mathbb{K}\to\mathcal{N}_{\mathfrak{G}}^{\pm\theta}/\mathbb{K},$$

where $\iota^{\pm\theta}:\mathcal{N}_{\mathfrak{g}}^{\pm\theta}/K\to\mathcal{N}_{\mathfrak{G}}^{\pm\theta}/\mathbb{K}$ is the natural orbit map. Steinberg-map compatibility conjecture. The orbit embedding is compatible with these maps, namely the diagram

$$\begin{CD} \mathfrak{X}/K @>{\Phi^{\pm\theta}}>> \mathcal{N}_{\mathfrak{g}}^{\pm\theta}/K \\ @V{\iota}VV @VV{\iota^{\pm\theta}}V \\ \mathbb{X}/\mathbb{K} @>{\mathbbl{\Phi}^{\pm\theta}}>> \mathcal{N}_{\mathfrak{G}}^{\pm\theta}/\mathbb{K} \end{CD}$$

is commutative.

This compatibility would relate the Steinberg maps for the smaller symmetric pair to those for the ambient pair. The source presents it as a natural expectation, and no resolution is supplied, so it remains open.