Correct. Thank you for clarifying. Just want to point out that this is different from your initial claim.
> housing to stop INCREASING in price
Land value tax would immediately start decreasing the housing price. You are correct that it would stay high until demand and supply rebalanced, but it would start decreasing.
In fact, you actually pointed out a second mechanism that could lead to decreasing prices in your comment: a decrease in demand. As you mentioned, that decrease in demand could come from a decline in population. However, it could also come from a decrease in that population's capability to match their existing level of spend. People can't spend more than they have so if that amount goes down then prices would also have to go down as a result.
In fact, prices for housing are far more sensitive to changes in demand than for normal goods. Land supply supply is fixed and cannot increase. But likewise, it cannot decrease either and this is what makes land value tax so amazing! Normally, a decrease in price would lead to a decrease in production, but with land, supply never changes so you get a larger decrease in price.
Economists refer to this property as having zero deadweight loss, and it is what makes land value tax the most efficient theoretical tax possible as was mentioned by Adam Smith over 100 years ago. Effectively, you can tax land up to its full theoretical rental value and supply will never change.
Land is an economic edge case. It's like a massive glitch in normal economic assumptions. And that glitch is literally the largest asset class globally and makes up the majority of bank lending. It needs special attention that it's not currently being given.
Practically speaking, land value tax advocates are not proposing full rental capture, but rather first replacing property taxes, and slowly increasing the tax rate while decreasing other taxes.