FV-width bounds for Feferman–Vaught reductions

Determine whether a theorem analogous to Theorem 6.1 of Dawar et al. can be proved to estimate the minimal Feferman–Vaught width of a first-order sentence.

Background

The paper explains that Feferman–Vaught reductions yield upper bounds on the DU-rank of classes definable in logical formalisms, but these bounds can be extremely large. The cited result of Dawar et al. shows that no elementary bound exists for a related preservation problem involving formula length and disjoint unions. The authors ask whether an analogous lower-bound or non-elementarity theorem can be established for the minimal FV-width of a formula.