Conjecture on descent of separated envelopes through universally closed birational morphisms

Let $T$ be an integral noetherian scheme. Suppose there exists a universally closed, surjective and birational morphism of integral noetherian schemes

$$p:T'\longrightarrow T,$$

and suppose that $T'$ has a separator. Then $T$ has a separated envelope

$$\varphi:T\longrightarrow F$$

in the sense of the definition used in the source.

Descent conjecture. Under these hypotheses, the separated envelope of $T'$ descends through $p$, so that $T$ possesses a separated envelope.

This conjecture concerns the existence of separated envelopes for integral noetherian schemes and whether that existence can be descended along universally closed birational surjective morphisms. The supplied text does not state a resolution, so its status remains open.