The conjecture on Morgan–Shalen accumulation of holonomy intersections

Let $S$ be the underlying surface, let $\mathcal{T}$ be its Teichmüller space, let $\mathcal{T}^\ast$ denote the corresponding space with the opposite orientation, and let $\chi$ be the character variety. For $X,Y\in\mathcal{T}\cup\mathcal{T}^\ast$, write $\chi_X$ and $\chi_Y$ for their holonomy varieties, and let the Morgan–Shalen boundary be the boundary of $\chi$ in its compactification by $\pi_1(S)$-actions on $\mathbb{R}$-trees. The Thurston boundary of the Teichmüller space is viewed as a subset of this boundary. Morgan–Shalen accumulation conjecture. For all distinct $X,Y\in\mathcal{T}\cup\mathcal{T}^\ast$, the accumulation set of $\chi_X\cap\chi_Y$ in the Morgan–Shalen boundary contains the Thurston boundary of the Teichmüller space in $\chi$. This would describe the asymptotic distribution of the infinitely many intersection points of holonomy varieties, extending the known infinitude and discreteness results toward a statement about their accumulation directions.