TITLE: Are etale morphisms "strongly formally etale"?
QUESTION [5 upvotes]: EDIT: Thanks to Harry Gindi and Anonymous below for insightful comments, I've refined the definitions here. Recall that a formally etale morphism of schemes $Y \to X$ is a morphism which has the unique right lifting property with respect to all nilpotent thickenings $Z \to W$. One salient feature of nilpotent thickenings is that they are universal homeomorphisms.
Definition: Say that a morphism of schemes $Y \to X$ is strongly formally etale if it has the unique right lifting property with respect to all universal homeomorphisms $Z \to W$. That is, for every commutative square as below, there exists a unique diagonal filler $W \to Y$, as indicated, making the two triangles commute.
$$\require{AMScd} \begin{CD} Z @>>> Y \\ @VVV \nearrow @VVV\\ W @>>> X \end{CD}$$
By definition, then, if $Y \to X$ is strongly formally etale, then $Y \to X$ is formally etale. The converse presumably does not hold. However, etale morphisms have an additional finiteness condition (etale = formally etale + locally of finite presentation) which makes me hope for an affirmative answer to the first question below:
Questions:

Let $Y \to X$ be an etale morphism. Then is $Y \to X$ strongly formally etale?
Does there exist standard terminology for "strongly formally etale"?
Is there a characterization of the class of morphisms which have the unique left lifting property with respect to all etale morphisms? How about (strongly) formally etale?

REPLY [6 votes]: This follows from the well-known "topological invariance of the étale site"; see e.g. Tag 04DZ.
Indeed, write $S' \to S$ for $Z \to W$, and consider the pullbacks of the étale map $Y \to X$ to étale maps $T \to S$ and $T' \to S$. We need to show that a section to $T' \to S'$ uniquely arises by pullback from a section of $T \to S$. This is a special case of Tag 0BTY.
Presumably the same should hold for weakly étale affine morphisms, since these are very close to ind-étale; see Tag 097Y. The only "topological invariance of the pro-étale site" I have seen is Lemma 5.4.2 is Bhatt–Scholze, which is only on the level of sheaves (rather than an actual equivalence of sites).