Um yeah then become the target of gossip and affect their work. Sounds like an 'awesome' idea. /s
Well, the girlfriend is already the object of gossip, but not associated with her boyfriend. If she's going to be gossiped about, wouldn't it be better to be gossiped about with her actual boyfriend?
I think you're right. Because if they both gotta keep working on the same job and department forever, keeping their relationship as secret won't help their love to develop further and will make the marriage impossible kind of. But well still there are gossips and seeing the boyfriend's behaviours, he would chicken out.
After reading the first 3 chapters, I hope there won't be any "fake relationship" just to avoid more rumors. Instead, they should just announce that they are a couple.