Fix two concept names A and B as well as some role name r. Does the entailment {∀r.A v ∃r.A} |= ∀r.B v ∃r.B hold true?

Yes, the entailment {∀r.A v ∃r.A} |= ∀r.B v ∃r.B holds true.

The left-hand side of the entailment, ∀r.A v ∃r.A, states that for any role r, either A holds for all instances of r or there exists at least one instance of r where A holds.

The right-hand side of the entailment, ∀r.B v ∃r.B, states that for any role r, either B holds for all instances of r or there exists at least one instance of r where B holds.

Since A and B are fixed concept names, the truth of either A or B is not dependent on r. Therefore, if A holds for all instances of r, then B must also hold for all instances of r, and if there exists at least one instance of r where A holds, then there must also exist at least one instance of r where B holds. Hence, the entailment holds true.