What type of dynamics arise in E₀-dilations of commuting quantum Markov semigroups?

Let H be a separable Hilbert space, and let φ and θ be two strongly commuting CP₀-semigroups on B(H). In a previous paper we constructed a Hilbert space K ⊇ H and two (strongly) commuting E ₀-semigroups α and β such that φ_s ̂ θ_t (P_H AP_H) =P_H α_ŝ β_t(A)P_H for all s, t < 0 and all A ∈ B(K). In this note we prove that if φ is not an automorphism semigroup, then the semigroup α (given by the above mentioned construction) is cocycle conjugate to the minimal *-endomorphic dilation of φ, and that if φ is an automorphism semigroup, then α is also an automorphism semigroup. In particular, we conclude that if φ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional), then α is a type I E₀-semigroup.