Simultaneously bounded and dense orbits for two diagonal flows

Let $d\geq 3$, let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, and let $v^1\ne v^2$ be two nonzero vectors in the diagonal subalgebra of $\mathfrak{sl}_d(\mathbb{R})$. Write $g_t^i=\exp(tv^i)$ for $i=1,2$ and $t\in\mathbb{R}$. Ray conjecture. The set

$$B(\{g_t^1\}_{t\geq 0})\cap D(\{g_t^2\}_{t\geq 0})$$

has full Hausdorff dimension in $X_d$. This extends the known $\mathrm{SL}_3$ result and is presented as an open strengthening of the paper's simultaneous bounded-and-dense orbit results.