Time evolution of density matrix for an interacting bipartite Hamiltonian

Thanks for looking into this question and for your patience since I am just learning all of this stuff. I have a bi-partite state (density matrices with trace 1 and positive):

$$
rho_{AB} = rho_A otimes rho_B,
$$

in the presence of an interacting Hamiltonian:

$$ mathcal{H}= hat{A} otimes hat{B},$$

where $hat{A}, rho_{A} in mathcal{B(H_A)}$ as well as $hat{B}, rho_{B} in mathcal{B(H_B)}$ where $mathcal{B(H)}$ denotes bounded operators on the Hilbert space $mathcal{H}$ . $rho_B$ may or may not be a mixed state of the form $rho_B = sum_i a_i |i_B rangle langle i_B| $. The question is if the time evolution of $rho_{AB}$ is of the following form (and how can I prove this?) :

$$rho_{AB}(t) = e^{it hat{A}} rho_A e^{-it hat{A}} , , otimes e^{it hat{B}} rho_B , e^{-it hat{B}} $$

Is this in general true? Do things change if the density matrix is now tripartite ie. $rho_{ABC} = rho_A otimes rho_B otimes rho_C$.