Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group G is the minimal natural number n such that every subgroup of G can be generated by n elements (or fewer).

This invariant has been studied extensively for various families of groups. I am interested in the family of finite simple groups and I have been unable to find and relevant information in the literature.

Question 1: Are there only finitely-many finite simple, non-abelian groups G of a given subgroup rank n?

Some relatively straight-forward comments and reductions:

It is not too difficult to show that there are only finitely-many alternating groups of subgroup rank at most n (by explicitly constructing elementary-abelian subgroups of a certain subgroup rank). There are also only finitely-many sporadic groups, according to the classification. These observations reduce the above question to finite simple groups of Lie-type.

Question 2: Are there only finitely-many finite simple groups G of Lie-type with given subgroup rank n?

It is again not too difficult to show that the “field rank” of G is bounded from above by a function of n (by looking at the natural homomorphism from the field to the root subgroups). It is also possible to show that the Lie-rank of G is bounded from above by a function of n. These observations further reduce question 2 to bounding the defining characteristic of the simple group of Lie-type by some function that depends only on the subgroup rank n. Unfortunately, I do not have any good intuition to determine whether the latter statement is true or not.

I hope both questions have a positive answer because that would give us a nice property about the FSG. But I suspect we can prove the answers to be “no” by simply making some judicious choice for the Lie-type, field-rank, and Lie-rank and by then looking at the structure of the Sylow-subgroups of G, as the characteristic goes through the different primes.