Housing investment would be 'housing starts' as in new houses being built. I doubt that prices will be going down if there are more people and the same number of houses. Part of the market for existing houses is sensitive to mortgages. All this means is housing will actually cost more in the future on a total cost of ownership basis. If you're a renter with a ton of cash, this is good, but for literally all other stakeholders in the market is it ultimately not good.
Sticker price for houses will fall, so if you're a cash buyer you get a discount. Monthly prices won't, so if you're a mortgage buyer you're paying more to the bank and less to yourself.
My understanding is as interest rates go up prices need to come down as people only have $X/month to allocate towards housing. EX: If you have $100 to put towards housing per month you can put $90 towards the house itself and $10 towards interest on the loan in low interest environments, but only $80 towards the house and $20 in high interest environments. If people can only put $80 per month towards the house you're selling that ultimately means the house price can't be as high as when people are able to put $90 per month towards the house.
For the renter with a lot of cash this means you can come out ahead if you're able to minimize the loan or outright purchase in cash a house. The renter with a lot of cash gets to benefit from the lower prices from higher interest rates while minimizing the downsides of higher interest rates.
