The first thing that he wished to settle was to speak with the Synthesis AI Administration, thus when everyone had settled themselves down for the day, he asked Lynn to follow him back to his room as they spoke about business. To be honest, she was expecting a personal scolding from him since the Crystal Dragon Mechanoid could have been brought down at that point in time, but they did not.
But she had no idea that Jin was going to do something more 'boyfriend' like by asking her to sit beside him and look through the options to discuss with the System before their proposal towards the Synthesis AI Administration. For someone who was terrible in expressing their emotions, this was probably the best roundabout way to ask for affection though Lynn positively accepted Jin's awkward way of presenting himself to her.
