20.70 (2022)

As a strengthening of the Burnside restriction, for every pair $(k, n)$ of positive integers, let a group $G$ satisfy condition $C_{k,n}$ if every $k$-generated subgroup of $G$ is finite of order at most $n$.
$\qquad$ a) Does there exist $k \geqslant 2$ such that for any $n$ all groups with condition $C_{k,n}$ are locally finite?
$\qquad$ b) In particular, is it true that for any $n$ the condition $C_{2,n}$ implies local finiteness?
$\qquad$ c) Find possibly more pairs $(k, n)$ for which groups with condition $C_{k,n}$ are locally finite. (For example, all groups with condition $C_{2,20}$ are metabelian, and therefore locally finite.)