This claim is largely opinion based, therefore unable to be falsified or proven to be true. Weight is subjective - there are different viewpoints on weight based on culture, geographic location, and personal beliefs. While there can be claims made about what are "healthy diets" or "effective weight loss methods", those would be more in the realm of a claim that could be called true or false.
"Happy weight", "Relationship weight", or "love weight" are all names given to the phenomena of couples gaining weight upon starting a relationship together. In the Washington Post's piece "Relationship weight gain is real — and can be a sign of happiness", they delved into studies being done at the University of North Carolina at Chapel Hill. Said studies, explained by Penny Gordon Larsen, a professor of global nutrition, have been difficult to conduct due to many different factors. These factors may speak to why your original claim is difficult to confirm. She went on to explain that they run into a lot of insufficient data during their attempts to study the phenomena, including only one partner participating, estimated pounds from participants, and the researchers only being able to conduct research after couples have already started dating. The window of time that the "relationship weight" can manifest itself is completely variable without a set time, weight, or indication. Professor Gordon Larsen went on to say that "happy weight" is entirely dependent on one's own perception of weight and weight gain.
Unhealthy eating habits are subjective in this instance too, different from person to person. It is true that the commonly thought theory of why "happy weight" is a phenomena is that people in secure relationships may feel less pressure to fit the standards that they once upheld for themselves while in the dating scene, or what they often call the "mating market model." Once again, these phenomenas are all perception based which make them unable to be confirmed. Still, the amount of variables that go into this phenomena are noted by researchers as they attempt to quantify what is happening in these instances.
Washington Post source:
https://www.washingtonpost.com/wellness/2022/08/09/relationship-weight-gain-reasons-research/