The Quillen equivalence conjecture for Segal categories and quasi-categories

Let $PrCat$ be the model category of Segal precategories, let $QCat$ be the model category of quasi-categories, and let $\phi:PrCat\to QCat$ and $\psi:QCat\to PrCat$ be the adjoint functors described in the source, with $\psi$ left adjoint to $\phi$. Quillen equivalence conjecture. The adjunction

$$\psi:QCat\longrightarrow PrCat\qquad QCat\longleftarrow PrCat:\phi$$

is a Quillen equivalence. This is one of the source's conjectural comparison statements among homotopy theories of higher categories; the surrounding discussion places it alongside comparisons with complete Segal spaces and simplicial categories.