Exactly. Because they hold monopoly pricing power over tenants they could in theory increase the rents 10 times even today. They don't do that, because better strategy for a parasite is to maximize profit and not to kill the host.
Now, if you abolish zoning or rezone places in addition to a land value tax, it could work as the investor could build more housing for the same land thus splitting the cost over multiple renters which I'm sure is the intended affect but a land value tax won't work without changing zoning laws and at the end of the day, it's all passed onto renters.
That logic only works in a non-competitive market. The reason they don't raise rates is obvious: There are other landlords around and any one landlord raising rates would lose business to the others. If you raise costs for all landlords, however—for example, by imposing a LVT—then there is no risk of defection. Landlords either raise their rates or lose money with every rent payment. Any that don't raise their rates will go out of business to be replaced with others who do. And if you use price controls to prevent them from raising rates despite losing money then they would go out of business without being replaced, leaving individuals and businesses to buy their own property outright rather than renting, in which case they will be paying the LVT along with all the other costs formerly handled by the landlord.
TL;DR: Don't assume that the market equilibrium price is fixed and independent of cost!
City rent is an aggregate monopoly - there is not enough supply of housing to have effect on downward pressure for rents. Demand is overwhelming supply.
Landlords' rent prices are not driven by competition but determined by what the market can bear.
Since no land can be created or destroyed, the supply-demand curve is fixed. This means that the land tax cannot be passed onto the renter, because the market conditions haven't changed. The rent is not calculated based on the landlord's costs plus some extra, but instead is whatever the market equilibrium price is.
Another way to think about it: if landlords have the market power to raise rents, why wouldn't they just do it now, rather than wait until an LVT is passed?