Best-of-both-worlds quadratic decomposition without L1 error

Establish whether there exists, even non-constructively, a structure–versus–randomness decomposition for every 1-bounded function f: F2^n → C and ε > 0 of the form f = ∑_{i=1}^r c_i (−1)^{p_i(·)} + g, with r = O(1/ε) and ∥g∥_{U^3} ≤ ε, that eliminates the additional L1 error term present in existing results while keeping the number of quadratic phases polynomial in 1/ε.

Background

The paper gives an efficient quadratic decomposition algorithm that produces a sum of O(1/ε) quadratic phases plus two error terms: one with small U^3 norm and another with small L1 norm. They also note that using alternative machinery can remove the L1 error at the cost of increasing the number of quadratic phases to exp(1/ε).

The open problem is whether a decomposition can simultaneously achieve both desiderata: O(1/ε) many quadratic terms and no L1 error term (i.e., only a U^3-small remainder), even ignoring algorithmic efficiency.