Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the inclusion $\operatorname{Kan} \hookrightarrow \operatorname{Cat}_\infty$. The following question came up in a discussion:

Suppose $D$ is an inversely-directed poset, and suppose $F:D\to \operatorname{Cat}_\infty$ is an injectively fibrant diagram of $\infty$-categories (wrt the Joyal model structure). Then is it true that the canonical map $\operatorname{Ex}^\infty(\lim F) \to \operatorname{lim}\operatorname{Ex}^\infty(F)$ is a weak homotopy equivalence (perhaps allowing for all of the limits in question to be appropriately homotopical)?

If not, is it true if $F(d)$ is isomorphic to the nerve of a poset for all $d\in D$?