Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure on $\mathrm{Set}_\Delta$ to define this). It has been proved that the $(\infty,1)$-category 2-$\mathrm{Cat}$ of $(\infty,2)$-categories is Cartesian closed. The category of bisimplicial sets $\mathrm{Fun}(\Delta^{op}, \mathrm{Set}_\Delta)$ is also Cartesian closed.

My question is:

Does $Y^X$ in 2-$\mathrm{Cat}$ coincide with $Y^X$ in $\mathrm{Fun}(\Delta^{op}, \mathrm{Set}_\Delta)$, i.e. is the latter one fibrant?

If not, is there a reference about the relation between the two (more specific than saying the first $Y^X$ is a fibrant replacement of the latter $Y^X$)?