Is the capacity achieving input of a discrete memoryless channel unique?

Question

Consider a classical discrete memoryless channel (DMC). Let $p$ be an input probability distribution and $Q$ be the channel's transition matrix. $q = Qp$ is a valid output probability distribution. The components of $p$ and $q$ are given as $p_i$ and $q_i$ respectively.

The capacity of this channel is given by maximizing the mutual information.

$$I = \max_{p}\left[\sum_ip_i\left(\sum_{j}Q_{ij}\log(Q_{ij})\right) - \sum_{j}q_j\log q_j\right]$$

But, we know (Theorem 2.7.4 in Cover and Thomas, 2006) that $I(p)$ is concave in $p$.

If $I(p)$ is not simply a linear function (i.e. $q = Qp$ is not a trivial relationship) and is differentiable everywhere (seems like a reasonable assumption), does this not imply that the capacity achieving $p$ is unique?