  In this paper, we investigate the Kirkwood-Dirac nonclassicality and
uncertainty diagram based on discrete Fourier transform (DFT) in a $d$
dimensional system. The uncertainty diagram of complete incompatibility bases
$\mathcal {A},\mathcal {B}$ are characterized by De Bi\`{e}vre [arXiv:
2207.07451]. We show that for the uncertainty diagram of the DFT matrix which
is a transition matrix from basis $\mathcal {A}$ to basis $\mathcal {B}$, there
is no ``hole" in the region of the $(n_{\mathcal {A}}, n_{\mathcal {B}})$-plane
above and on the line $n_{\mathcal {A}}+n_{\mathcal {B}}\geq d+1$, whether the
bases $\mathcal {A},\mathcal {B}$ are not complete incompatible bases or not.
Then we present that the KD nonclassicality of a state based on the DFT matrix
can be completely characterized by using the support uncertainty relation
$n_{\mathcal {A}}(\psi)n_{\mathcal {B}}(\psi)\geq d$, where $n_{\mathcal
{A}}(\psi)$ and $n_{\mathcal {B}}(\psi)$ count the number of nonvanishing
coefficients in the basis $\mathcal {A}$ and $\mathcal {B}$ representations,
respectively. That is, a state $|\psi\rangle$ is KD nonclassical if and only if
$n_{\mathcal {A}}(\psi)n_{\mathcal {B}}(\psi)> d$, whenever $d$ is prime or
not. That gives a positive answer to the conjecture in [Phys. Rev. Lett.
\textbf{127}, 190404 (2021)].
