Depth⁺ and length⁺ of boolean algebras

Suppose that κ = cf(κ), λ > cf(λ) = κ⁺ and λ = λ^κ. We prove that there exist a sequence hB_i : i < κi of Boolean algebras and an ultrafilter D over κ so that λ = Q Depth⁺(B_i)/D < Depth⁺(Q B_i/D) = λ⁺. An i<κ i<κ identical result holds also for Length⁺. The proof is carried in ZFC, and it holds even above large cardinals.