  For each non-constant $q$ in the set of $n$-variable Boolean functions, the
{\em $q$-transform} of a Boolean function $f$ is related to the Hamming
distances from $f$ to the functions obtainable from $q$ by nonsingular linear
change of basis. Klapper conjectured that no Boolean function exists with its
$q$-transform coefficients equal to $\pm 2^{n/2}$ (such function is called
$q$-bent). In our early work, we only gave partial results to confirm this
conjecture for small $n$. Here we prove thoroughly that the conjecture is true
by investigating the nonexistence of the partial difference sets in Abelian
groups with special parameters. We also introduce a new family of functions
called almost $q$-bent functions, which are close to $q$-bentness.
