Yeah, sorry, I misunderstood the stats.
Let me try again. Stats from AI, because I can't be arsed looking properly. 830,000 Londoners live in social housing. Of those, around 390,000 are foreign-born, and 440,000 UK-born. 3.7 million Londoners are foreign-born, which means just over 10% are in social housing. 5.2 million Londoners are UK-born, meaning around 8.5% are in social housing. So foreign-born people are slightly more likely to end up in social housing, and this is ignoring the fact that a lot of foreign born people are not eligible, which suggests that those who are eligible are quite a bit more likely to end up in social housing than the native population. Obviously all of that is just from a quick Google search and the AI answers, so I'm not going to put a huge amount of stock into it.
It also doesn't give any idea of who these people are. Obviously the anti-immigrant narrative is that people are coming over and immediately handed a free house. In reality, we might be talking about pensioners who came over in their early 20s and spent their whole life living and working in the UK.