The article you cited doesn't support that assertion. Its thesis is that upzoning alone — i.e. relaxing regulations such that it is legal to build higher-density housing, without further interventions — may not be sufficient to create enough vacancies to lower rents.
The cited article alone simply admits that upzoning won't result in cheaper housing. Because the market is broken (and only socialized housing can fix it), but we must do upzoning anyway.
That article also doesn't support your assertion. For example, they specifically call out parking minimums and minimum lot sizes (both density-lowering regulations) as major drivers of high housing costs.
The article you cited doesn't support that assertion. Its thesis is that upzoning alone — i.e. relaxing regulations such that it is legal to build higher-density housing, without further interventions — may not be sufficient to create enough vacancies to lower rents.