  It has been asked by different authors whether the two classes of
Schatten-$p$-norm-based functionals
$C_p(\rho)=\min_{\sigma\in\mathcal{I}}||\rho-\sigma||_p$ and $
\tilde{C}_p(\rho)= \|\rho-\Delta\rho\|_{p}$ with $p\geq 1$ are valid coherence
measures under incoherent operations, strictly incoherent operations, and
genuinely incoherent operations, respectively, where $\mathcal{I}$ is the set
of incoherent states and $\Delta\rho$ is the diagonal part of density operator
$\rho$. Of these questions, all we know is that $C_p(\rho)$ is not a valid
coherence measure under incoherent operations and strictly incoherent
operations, but all other aspects remain open. In this paper, we prove that (1)
$\tilde{C}_1(\rho)$ is a valid coherence measure under both strictly incoherent
operations and genuinely incoherent operations but not a valid coherence
measure under incoherent operations, (2) $C_1(\rho)$ is not a valid coherence
measure even under genuinely incoherent operations, and (3) neither
${C}_{p>1}(\rho)$ nor $\tilde{C}_{p>1}(\rho)$ is a valid coherence measure
under any of the three sets of operations. This paper not only provides a
thorough examination on the validity of taking $C_p(\rho)$ and
$\tilde{C}_p(\rho)$ as coherence measures, but also finds an example that
fulfills the monotonicity under strictly incoherent operations but violates it
under incoherent operations.
