Indeed, it seems to be a question of soundness. More precisely, the condition that $P_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every ${\cal K}$-limit preserving presheaf $F:{\cal C}^{op} \to {\rm Set}$ is an ${\cal I}$-colimit of representables, or, in this case, a ${\cal K}$-filtered colimit of representables. Looking in this paper we see that this indeed holds whenever ${\cal K}$ is sound (theorem 2.4), and that furthermore, this being the case appears to be the main motivation for the definition of soundness. For example, it seems to me that $P_{\cal I}({\cal C})$ doesn't contain the terminal presheaf in the non-sound case described in Example 2.3 (vi) of loc.cit.

For the 1-categorical case, it seems to be indeed a question of soundness. More precisely, the condition that $P_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every ${\cal K}$-limit preserving presheaf $F:{\cal C}^{op} \to {\rm Set}$ is an ${\cal I}$-colimit of representables, or, in this case, a ${\cal K}$-filtered colimit of representables. Looking in this paper we see that this indeed holds whenever ${\cal K}$ is sound (theorem 2.4), and that furthermore, this being the case appears to be the main motivation for the definition of soundness. For example, it seems to me that $P_{\cal I}({\cal C})$ doesn't contain the terminal presheaf in the non-sound case described in Example 2.3 (vi) of loc.cit.

I would say that it's quite likely that the $\infty$-categorical case will behave similarly, once one defines soundness appropriately (maybe replace "connected" with "weakly contractible" in Definition 2.2 of loc.cit)?