  A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16)
shows that any bounded quadratic polynomial can be computed exactly in
expectation by a 1-query algorithm up to a universal multiplicative factor
related to the famous Grothendieck constant. A natural question posed there
asks if bounded quartic polynomials can be approximated by $2$-query quantum
algorithms. Arunachalam, Palazuelos and the first author showed that there is
no direct analogue of the result of Aaronson et al. in this case. We improve on
this result in the following ways: First, we point out and fix a small error in
the construction that has to do with a translation from cubic to quartic
polynomials. Second, we give a completely explicit example based on techniques
from additive combinatorics. Third, we show that the result still holds when we
allow for a small additive error. For this, we apply an SDP characterization of
Gribling and Laurent (QIP'19) for the completely-bounded approximate degree.
