A friend of mine is in exactly the same situation now. His wife is a teacher (synonym for infinite income & tons of benefits in Germany, surely you know). And still he has to pay the mortgage because the kids must be given a chance to finish the elementary school, because otherwise she‘d need to move out. He can’t sell his part of the house because of the kids, but he can’t live there either 🤷🏻♂️🤦♂️
To be honest what Germany does to fathers is embarrassing in many ways. Luckily, he found another job with a decent salary to not care about this shit.