The challenge is that LVT as advocated here pushes for annual updates of the tax based on available information. That easily can move rates up (and down) by “drive people out” amounts in 5-10 years in urban areas.
Of course any government can jack up taxes, and some may use LVT as an excuse. That's already a problem. But the _ideal_ LVT replaces property tax. So I'm skeptical that LVT will be a large exacerbating factor.
The real problem is that gentrification drives up demand, and that's baked into the premise - and necessarily precedes any LVT uprating. I expect LVT's contribution to the problem to be marginal. More data needed.