Catatonic immobility is associated with psychiatric conditions such as schizophrenia. it is characterized by disturbances in motor (muscular movement) behavior with either psychological or physiological basis. It can include periods in which the individual moves very little and does not respond to instructions. The individual holds rigid poses for a considerable length of time. They may maintain this position for hours, days, weeks or even months.
I think this would be an example of companionate love because it seems it is based more on a friendship. Companionate love can form in non-romantic relationships, and this relationship does not seem very romantic.