In theory yes, but this tax was introduced in recent memory and the rate of rent increases that year was not noticeably different to the rate of rent increases

In theory yes, but this tax was introduced in recent memory and the rate of rent increases that year was not noticeably different to the rate of rent increases in other years, as the prices are limited (well "limited") by the ability of tenants to pay, rather than being driven by costs. And the most expensive areas have rent control anyway.