Tightness of the 16·rational-degree^4 upper bound on decision-tree complexity

Determine whether there exists a Boolean function f for which the upper bound D(f) ≤ 16·rdeg(f)^4 is tight, by identifying an explicit function that achieves equality up to constant factors or proving that such a function does not exist.

Background

The authors show D(f) ≤ 2·ndeg(f)^2·ndeg(¬f)^2 ≤ 2·rdeg(f)^4 and D(f) ≤ 4·sdeg(f)^2·rdeg(f)^2 ≤ 16·rdeg(f)^4. They note that the 2·rdeg(f)^4 bound is tight for two-bit parity.

However, they explicitly state that they do not know any function for which the 16·rdeg(f)^4 bound is tight, leaving open the question of its tightness.