Relation between local nilpotency class of group and inner automorphism group

Statement
Suppose $$G$$ is a group and $$m,c$$ are positive integers such that the $$m$$-local nilpotency class of $$G$$ is at most $$c$$. Suppose $$Z(G)$$ is the center of $$G$$, so that the inner automorphism group of $$G$$, denoted $$\operatorname{Inn}(G)$$, is isomorphic to $$G/Z(G)$$. Then, the $$(m - 1)$$-local nilpotency class of $$\operatorname{Inn}(G)$$ is at most $$c - 1$$.

Note that if we took $$m = \infty$$, this would simply tell us that the nilpotency class of the inner automorphism group is one less than that of the original group.

Related facts

 * Relation between local nilpotency class of group and derived subgroup
 * Relation between local nilpotency class of Lie ring and inner derivation ring