Gibbs' inequality is equivalent to:

\begin{equation} \sum_{i} \ln q_i^{p_i}-\ln p_i^{p_i} \geq 0 \end{equation}

Now, a friend of mine suggested that:

\begin{equation} \sum_{i} \ln q_i^{p_i}-\ln p_i^{p_i} \geq 0 \implies \sum_{i} q_i^{p_i}-p_i^{p_i} \geq 0 \end{equation}

Right now I doubt this is true but I can't think of a counter-example. I'm not particularly good at manipulating inequalities myself.