LVT vs PVT in short term?

I understand that a land value tax in theory, keeps property prices and rents the same, and the incidence of the tax would be eaten by the landowners due to fixed supply, the apparently perfect tax. Although to me it seems counterintuitive, I'll go with it.

My question is, would the same be true for a property value tax? Since in the immediate term, new housing stock can't be a factor since they take years to build, would they not be functionally equivalent initially on average? Would the ratio of rental properties to owned properties be different in each case?

This is only in a short time frame, I understand that past the initial period, an LVT would incentivize development whereas a PVT would not.