In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:

For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index subgroup of rank $2$.

He has probably also conjectured that the rank $2$ subgroups form a basis for the profinite topology (see Problem 1 of Section 4 in his work).

A 2010 work of Long and Reid ("Small subgroups of $\mathrm{SL}_3(\mathbb{Z})$") confirmed the conjecture for $n=3$.

I am interested in the current status of this conjecture.

More concretely, denote by $q(n)$ the least integer for which there are infinitely many finite index subgroups of $\mathrm{SL}_n(\mathbb{Z})$ of rank at most $q(n)$.

What is the best known bound for $q(n)$ ?

Sury and Venkataramana ("GENERATORS FOR ALL PRINCIPAL CONGRUENCE SUBGROUPS OF $\mathrm{SL}_n(\mathbb{Z})$ WITH $n \geq 3$") show that $q(n) \leq n^4$ by presenting generating sets for principal congruence subgroups.

Both proofs and references to related works will be highly appreciated.

Here the rank of a group is the smallest cardinality of a generating set.

A related work is that of Sharma and Venkataramana ("Generations for arithmetic groups") where it is shown that any noncocompact irreducible lattice in a higher rank real semi-simple Lie group contains a subgroup of finite index with rank at most $3$.