Don't know if there is a japanese equivalent, but they would be "severing their tie"
The Japanese equivalent is alluded to in the ending bubble: she doesn't want to cut their
red cloth(ribbon)
of fate.
It's also called a red string of fate sometimes, or a red thread of marriage. It's an East Asian belief which symbolizes a fated connection between two soulmates/fated lovers.
Cutting it (as in cutting off an actual red thread of fate) would be equivalent to severing your connection with your soulmate and forever losing the chance to end up with them (or at the very least making it much less likely).