Existence of marked $(\infty,2)$-colimits

Let $\mathcal{C}$ be an $(\infty,2)$-category, let $X$ be a marked simplicial set, and let $F:X\to\mathcal{C}$ be a functor. A marked $(\infty,2)$-colimit theory should categorify the strict 2-categorical theory and assign to $F$ a marked $(\infty,2)$-colimit cone over $F$.

Existence conjecture. There is a theory of marked $(\infty,2)$-colimits in every $(\infty,2)$-category, with input and output as above.

The conjecture proposes the required $(\infty,2)$-categorical extension of marked colimits from strict 2-categories; the paper notes that constructing marked cocones and proving the necessary fibrancy and functoriality are technical difficulties.