Lafont's bounded-cohomology surjectivity conjecture for higher-rank symmetric spaces

Let $G$ be a connected noncompact simple Lie group and $X$ an irreducible symmetric space. Given a geodesic $\gamma$ in $X$, let $F(\gamma)$ be the union of all geodesics in $X$ parallel to $\gamma$, and set

$$t_X=\max_{\gamma}\dim(F(\gamma)).$$

Let $\Gamma$ be a cocompact lattice in $G$. Lafont's surjectivity conjecture. The comparison map

$$H^d_b(\Gamma,\mathbb R)\longrightarrow H^d(\Gamma,\mathbb R)$$

is surjective for $d\geq t_X+1$. The conjecture concerns the degree range in which bounded cohomology detects ordinary cohomology for lattices in higher-rank groups. The source reports evidence in rank $2$, where surjectivity is established in degrees at least $\max\{6,t_X+2\}$, leaving in most cases only the degree $d=t_X+1$ unresolved.