Ending rent control also should lead to lower prices in non rent controlled housing according to supply and demand.
The sequence of events that lead to a rise in prices are similar - that is there is evidence that more affluent people moving in to a local area can result in rents going up for others in that same area.
A good example is Jerry Brown's 10K Project — his plan to attract 10,000 people to live in downtown Oakland by 2001, which was the cornerstone of his tenure as mayor. The idea was that 10k affluent people would create a positive feedback loop of gentrification. As someone living in the area in the 90s-2000s, I can say it "worked" but at the expense of the working class as Oakland began gentrifying rapidly by 2010.
Raising the price of one good usually causes the price of substitute goods to go up. For example, if the price of wheat goes up because of import duties, the price of rice will also go up as people switch from wheat to rice.
In the rent control example, many people held on to RC apartments because they were a great deal, even though they weren't where they wanted to live. After RC ended, they moved to apartments they preferred (since there was no longer a big price differential), increasing demand (and thus price) for the originally non-RC apartments.
The sequence of events that lead to a rise in prices are similar - that is there is evidence that more affluent people moving in to a local area can result in rents going up for others in that same area.