Earlier quoted context omitted.
https://en.wikipedia.org/wiki/Land_value_tax#Efficiency A landlord's decision to improve or not improve the property only depends on 1) the cost of the improvements and 2) the expected revenue stream from that improvement, neither of which are affected by an LVT, since the LVT only cares about the unimproved value of the lot.
I can't follow the technical arguments you're making.
There are only two things you care about:
1. How much money does that improvement bring in?
2. How much does that improvement cost?
Neither of those two inputs are affected by a land value tax. Therefore the landlord must reach the same decision, regardless of whether there is an LVT or not, and so LVTs have no effect on whether landlords improve their property.
You might think that they'll be motivated to improve the lot in order to be able to pay the LVT, but that rationally doesn't factor in to the decision to improve the land or not. It only affects the price of the land (Imposing an LVT makes land prices fall by the discounted future cost of the tax).
And not to be rude, but it seems strange to jump into discussions recommending LVTs as a solution to an economic problem without first understanding basic economic theory.