Simultaneously bounded and dense orbits for countably many diagonal flows

Let $d\geq 3$, let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, and let $I_m,J_n$ be index sets. Choose nonzero vectors $\{v^{1,i}\}_{i\in I_m}\cup\{v^{2,j}\}_{j\in J_n}$ in the diagonal subalgebra of $\mathfrak{sl}_d(\mathbb{R})$ such that $v^{1,i}\ne v^{2,j}$ for every $(i,j)\in I_m\times J_n$. Define $g_t^{1,i}=\exp(tv^{1,i})$ and $g_t^{2,j}=\exp(tv^{2,j})$. Countable simultaneous-orbit conjecture. The set

$$\bigcap_{i\in I_m}B(\{g_t^{1,i}\}_{t\geq 0})\cap\bigcap_{j\in J_n}D(\{g_t^{2,j}\}_{t\geq 0})$$

has full Hausdorff dimension in $X_d$. The cases with only bounded or only dense conditions are known in stronger forms, while the simultaneous general case is proposed as an open conjecture.