Characterization of the universal isometry group of q-map spaces

Let $\overline{N}\cong \mathcal{H}\times L$ be the q-map space, with $g_{L}(p)$ the left-invariant metric on the fiber $L_p\cong L$ parametrized by $p\in\mathcal{H}$. Let $G$ be the connected Lie subgroup of $\mathrm{Iso}_0(\overline{N},g_{\overline{N}})$ whose Lie algebra is

$$\mathfrak{g}=\mathbb{R}D\ltimes\bigl(\mathfrak{sl}_2(\mathbb{R})^{\mathrm{op}}\ltimes(\mathbb{R}^n\ltimes\mathfrak{h})\bigr).$$

For each $p\in\mathcal{H}$, let $\mathrm{Iso}_0(L,g_L(p))$ be the connected component of the identity of the isometry group of $(L,g_L(p))$. Universal isometry group characterization. Then

$$G=\bigcap_{p\in\mathcal{H}}\mathrm{Iso}_0(L,g_L(p)).$$

This conjecture proposes that the universal isometry group generated by the $L$-action and S-duality is exactly the subgroup acting by isometries on every fiber metric $g_L(p)$. The supplied text does not state whether the conjecture has been resolved.