Confined water chains have been shown to display interesting properties, such as phase transitions (PTs), and these properties could be leveraged to create quantum devices. More specifically, a temperature-dependent quasi-PT was found experimentally for water-filled (6, 5) carbon nanotubes (CNT). Simulations have attempted to explain this PT, suggesting that it was caused by an orientational order to disorder transition in the dipoles of the water chain. However, this work was done with classical simulations, and presented only a limited analysis of the mechanisms at play. To address this and obtain a conclusive PT mechanism, we present our quantum simulations of this system where emphasis has been placed on creating realistic experimental conditions. Analytically-defined caps are used to investigate the system at the equilibrium density and precomputed discretized potential energies and forces help accelerate path integral molecular dynamics simulations to reach ergodicity. A wide range of temperatures are simulated to accurately capture the whole transition and the system size is changed to help pinpoint the PT's thermodynamic limit. The tools and methodology developed for the comprehensive study of the water-filled CNT will naturally lend themselves to other confined water systems, and studying these could enable tuning of the currently observed quantum properties or uncover new ones.