Uniqueness conjecture for the 15-vertex triangulation of a non-orientable 4-manifold

Let M=(S^3\text{\times\hspace{-1.62ex}_\hspace{-.4ex}_\hspace{.7ex}}S^1)\#({\mathbb C}{\bf P}^{\,2})^{\# 5}, and let $\text{}^4\hspace{.3pt}15^{\,4}_{\,1}$ denote the specified combinatorial triangulation of $M$. A triangulation is vertex-minimal if it uses the minimum possible number of vertices among triangulations of the same manifold. Uniqueness conjecture. The combinatorial manifold $\text{}^4\hspace{.3pt}15^{\,4}_{\,1}$ is the unique vertex-minimal triangulation of $M$ with $15$ vertices. The paper identifies $\text{}^4\hspace{.3pt}15^{\,4}_{\,1}$ as a vertex-transitive triangulation of this manifold, but does not establish the asserted uniqueness.