Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition?

One way to get such a bound would be to give a lower bound for the dimension of a non-trivial representation (over $\mathbb Q$, or over $\mathbb C$ — I don't know if these are the same for these groups). Possibly such a bound is obvious from a good knowledge of Deligne–Lusztig theory but this is something i would like to avoid if possible.

My naïve guess would be $p^r$ (where $r$ is the rank), from the example of $\mathrm{PSL}_{r+1}(\mathbb F_p)$ where there is a subgroup of index roughly $p^r$ (the maximal parabolic with Levi component $\mathrm{GL}_r$) and it seems hard (impossible?) to come up with subgroups of lower index. For $r=1, 2$ one can actually prove that $p^r$ is the right order using a lower bound on the dimension of a nontrivial representation (proven by elementary means), I don't know whether this generalises to higher ranks.