In terms of milking LKY's legacy for his own political gain (as what happened in GE2015), the presence of 38 Oxley Road really does not have much impact.
Right now, the siblings are concentrating all their fire power on this issue but I'm pretty sure it goes deeper. LKY suspected his son was going to paint him as a national superhero and bask in his glory in order to secure the next election and surely must have felt uncomfortable. This was in fact raised by crown princess LWL some months ago.
I'd be surprised if 38 Oxley is the only conflict that they have between them.