The maximality conjecture for proper maps under double orthogonals

Let $Q$ be any class of morphisms of topological spaces, and write $Q^{lr}$ for its double Quillen orthogonal. The proper-map maximality conjecture. If

$$\{\text{proper maps}\}\subsetneq Q^{lr},$$

then

$$Q^{lr}=\{\text{all morphisms}\}.$$

The paper proves the analogous implication when $Q$ consists of maps of finite spaces, but leaves the assertion for arbitrary classes open.