More on regular reduced products

The authors show, by means of a finitary version □ _λ.D^fin of the combinatorial principle □_λ^b* of [7]. the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ, if M_i and N_i are elementarily equivalent models of a language of size <λ, then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ⁺ on ∏_i M_i/D and ∏_i N_i/D. If in addition 2^λ = λ⁺ and i < λ implies |M_i| + |N_i| ≤ λ⁺ this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in [2].