Rigidity of five-dimensional quasi-Einstein manifolds with constant scalar curvature

Let $(M^5,g)$ be a five-dimensional non-trivial simply-connected compact quasi-Einstein manifold with boundary. If $M$ has constant scalar $R$, Johnatan Costa, Ernani Ribeiro Jr, and Detang Zhou show that $R$ = $((m-5)k+20)\mathrm{/}(m-k+4)\lambda$ for some $k\in \{0,2,3,4\}$. Both cases of $k=0$ and $k=4$ are already classified. In this paper we will prove that the case $k=3$ is rigid.