On a problem about I-projections

The minimizer P* of the I-divergence D(P||Q) for P in a set ε defined by linear constraints is known to be mutually absolutely continuous with Q (P*≡Q) providing a P˜ in ε exists with P˜≡Q and D(P˜||Q)<∞. We ask when the existence of P¯ and P, both in ε, with P¯≡Q and D(P||Q)<∞ is already sufficient for P*≡Q. We give a positive answer for measures on a product space when ε is determined by prescribing the two marginals