Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial title proposed, that the former should be understood as a relative notion, constructions relative to a choice of base $\infty$-topos, while the latter involves the choice of a specific such base. This would suggest that a useful thing to investigate is the replacement in all things cohesive of the default base $\infty$-topos of $\infty$-groupoids by $Sh_{\infty}(ProFinSet)$. We might expect then that the package of constructions of cohesion in its various forms (infinitesimal, differential, singular (for orbifolds), elastic, solid, etc.) would be automatically available for this new base.

What I would like to gain a better sense of is how important such constructions are likely to be.

As a starter, then, have people considered the tangent $\infty$-topos, $T(Sh_{\infty}(ProFinSet))$, a case of infinitesimal cohesion over the new base? Presumably this involves some form of parameterised pyknotic/condensed spectra.

From the above-mentioned conversations, it appears that the $\infty$-topos of $\infty$-sheaves over the pro-étale site on all schemes over a separably closed field $k$ is cohesive over $Sh_{\infty}(ProFinSet)$. Then for any spectrum object in such a cohesive setting, the associated differential cohomology follows from the relevant differential cohomology hexagon. Is this likely to be important?

The article here covers all forms of cohesion.

Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial title proposed, that the former should be understood as a relative notion, constructions relative to a choice of base $\infty$-topos, while the latter involves the choice of a specific such base. This would suggest that a useful thing to investigate is the replacement in all things cohesive of the default base $\infty$-topos of $\infty$-groupoids by $Sh_{\infty}(ProFinSet)$. We might expect then that the package of constructions of cohesion in its various forms (infinitesimal, differential, singular (for orbifolds), elastic, solid, etc.) would be automatically available for this new base.

What I would like to gain a better sense of is how important such constructions are likely to be.

As a starter, then, have people considered the tangent $\infty$-topos, $T(Sh_{\infty}(ProFinSet))$, a case of infinitesimal cohesion over the new base? Presumably this involves some form of parameterised pyknotic/condensed spectra.

From the above-mentioned conversations, it appears that the $\infty$-topos of $\infty$-sheaves over the pro-étale site on all schemes over a separably closed field $k$ is cohesive over $Sh_{\infty}(ProFinSet)$. Then for any spectrum object in such a cohesive setting, the associated differential cohomology follows from the relevant differential cohomology hexagon. Is this likely to be important?

The article by Hisham Sati & Urs Schreiber, Proper Orbifold Cohomology, covers all forms of cohesion.