Marked colimits in the infinity-bicategory of infinity-categories

Let $\mathfrak{C}\!\operatorname{at}_{\infty}$ denote the $\infty$-bicategory of $\infty$-categories. For a marked simplicial set $X$, write $X^\flat$ and $X^\sharp$ for the corresponding minimally and maximally marked simplicial sets, and regard functors into $\operatorname{Cat}_\infty$ as functors into $\mathfrak{C}\!\operatorname{at}_{\infty}$.

Special-cases conjecture. In $\mathfrak{C}\!\operatorname{at}_{\infty}$, the marked $(\infty,2)$-colimit of $F:X^\flat\to\mathfrak{C}\!\operatorname{at}_{\infty}$ coincides with the lax $\infty$-colimit of the underlying functor of $F$, while the marked $(\infty,2)$-colimit of $F:X^\sharp\to\operatorname{Cat}_\infty\to\mathfrak{C}\!\operatorname{at}_{\infty}$ coincides with the $(\infty,1)$-colimit.

These expected identifications connect marked $(\infty,2)$-colimits with familiar lax and $(\infty,1)$-categorical colimits. They are presented as special cases suggesting the broader theory, whose existence and rigorous foundations remain under development.