  Mizuno provided 15 examples of generalized rank three Nahm sums with
symmetrizer $\mathrm{diag}(1,2,2)$ which are conjecturally modular. Using the
theory of Bailey pairs and some $q$-series techniques, we establish a number of
triple sum Rogers--Ramanujan type identities. These identities confirm the
modularity of all of Mizuno's examples except for two non-modular cases. We
show that the two exceptional cases of Nahm sums are sums of modular forms of
weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation
formulas for two vector-valued functions consisting of Nahm sums with
symmetrizers $\mathrm{diag}(1,1,2)$ and $\mathrm{diag}(1,2,2)$.
