  This work deals with the entropic regularization of the Brenier problem for
perfect incompressible fluids introduced by Arnaudon, Cruzeiro, L\'eonard and
Zambrini. We show that as in the original setting, there exists a scalar
pressure field which is interpreted as the Lagrange multiplier associated to
the incompressibility constraint. The proof goes through a reformulation of the
problem in PDE terms.
