\begin{align}\label{eq:lower-bound-time-ours-2}
\sup_{\boldsymbol{\tilde{\pi}} \in \widetilde{\Sigma}_K} \inf_{\boldsymbol{\lambda} \in \textrm{Alt}(\boldsymbol{\mu})} \sum_{a=1}^K \tilde{\pi}_a d(\mu_a, \lambda_a).
\end{align}
