  We give improved separations for the query complexity analogue of the
log-approximate-rank conjecture i.e. we show that there are a plethora of total
Boolean functions on $n$ input bits, each of which has approximate Fourier
sparsity at most $O(n^3)$ and randomized parity decision tree complexity
$\Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and
Sherif (JACM '20) both qualitatively (in terms of designing a large number of
examples) and quantitatively (improving the gap from quartic to cubic). We
leave open the problem of proving a randomized communication complexity lower
bound for XOR compositions of our examples. A linear lower bound would lead to
new and improved refutations of the log-approximate-rank conjecture. Moreover,
if any of these compositions had even a sub-linear cost randomized
communication protocol, it would demonstrate that randomized parity decision
tree complexity does not lift to randomized communication complexity in general
(with the XOR gadget).
