Which limits commute with all colimits in $T(h)$-local spectra?

Question

Consider the category $Sp_{T(h)}$ of $T(h)$-local spectra. Let $J, K$ be small $\infty$-categories. Recall that $J$-limits said to commute with $K$-colimits in $Sp_{T(h)}$ if, for all functors $F : J \times K \to Sp_{T(h)}$, the canonical map

$$\varinjlim_{k \in K} \varprojlim_{j \in J} F(j,k) \to \varprojlim_{j \in J} \varinjlim_{k \in K} F(j,k)$$

is an equivalence.

Question: For which small $\infty$-categories $J$ do $J$-limits commute with $K$-colimits in $Sp_{T(h)}$ for all small $\infty$-categories $K$?

There are some noteworthy examples of such $J$. In all cases, the reasoning is "$J$-limits can be re-expressed in terms colimits, which always commute with colimits by the "Fubini rule". (Indeed, such colimits are absolute, and general nonsense entails that when $J$ answers to the Question, there is always an argument of this form).

1. Idempotent splitting is a form of limit which commutes with colimits in $Sp_{T(h)}$ (simply because $Sp_{T(h)}$ is an $\infty$-category).

2. Finite limits commute with colimits in $Sp_{T(h)}$ (because $Sp_{T(h)}$ is stable).

3. Limits indexed by $\pi$-finite spaces commute with colimits in $Sp_{T(h)}$ (because $Sp_{T(h)}$ is $\infty$-ambidextrous).

There is also some mixing and matching which can occur between the above examples. But what does the general picture look like?