Mixed-structure compactification conjecture for rank-two Hitchin and maximal components

Let $S$ be a surface and let $G$ be a real Lie group of rank $2$. A $(\overline{\mathfrak{a}^{+}},k)$-mixed structure on $S$ is the structure whose dual $(\mathfrak{a},W)$-complex has a one-dimensional part given by trees dual to the laminar part and a two-dimensional part with a $1/k$-translation-surface structure, where $\overline{\mathfrak{a}^{+}}$ is a Cartan subalgebra of the Lie algebra of $G$ and $k$ depends on $G$. For a divergent sequence of Hitchin or maximal representations, the rescaled equivariant minimal surfaces should converge to a subcomplex of the corresponding Euclidean building. Mixed-structure compactification conjecture. The limiting complex $\Sigma_{\infty}$ is dual to a $(\overline{\mathfrak{a}^{+}},k)$-mixed structure. More precisely, if $G$ is a real split semisimple Lie group of rank $2$, then the boundary of $\mathrm{Hit}(S,G)$ can be identified with the space of projective classes of such mixed structures, with $(\mathfrak{a},k)=(A_{2},3)$ for $\mathrm{SL}(3,\mathbb{R})$, $(B_{2},4)$ for $\mathrm{Sp}(4,\mathbb{R})$, and $(G_{2},6)$ for $G_{2}^{\mathbb{R}}$. If $G$ is a real semisimple Lie group of Hermitian type and rank $2$, then the boundary of $\mathrm{Max}(S,G)$ can be identified with that space, with $(\mathfrak{a},k)=(A_{1}\times A_{1},2)$ for $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ and $(B_{2},4)$ for $\mathrm{SO}(2,n)$ with $n\geq 3$. The conjecture proposes a geometric description of the boundary of these character-variety components through limiting minimal surfaces in Euclidean buildings; its general resolution is not established in the supplied text.