The cofinality conjecture for marked infinity-2-functors

Let $\mathbfcal{C}^\dagger$ and $\mathbfcal{D}^\dagger$ be marked $(\infty,2)$-categories, and let $F:\mathbfcal{C}^\dagger\to\mathbfcal{D}^\dagger$ be a marked $(\infty,2)$-functor. For each $d\in\mathbfcal{D}$, let (\mathbfcal{C}_{d\!% \mathrel{\text{\rotatebox[origin=c]{65}{\vrule height 2\fontdimen22\textfont2 width 0pt\rightarrow}}} })^\dagger and (\mathbfcal{D}_{d\!% \mathrel{\text{\rotatebox[origin=c]{65}{\vrule height 2\fontdimen22\textfont2 width 0pt\rightarrow}}} })^\dagger denote the marked $(\infty,2)$ slice categories, and let $L_{\mathcal{W}}$ denote their $(\infty,1)$-localization.

The cofinality conjecture. Precomposition with $F$ preserves marked $(\infty,2)$-colimits if and only if, for every object $d\in\mathbfcal{D}$: (1) there are $c\in\mathbfcal{C}$ and a morphism $d\to F(c)$ initial in the $(\infty,1)$-localizations of both slices; (2) every marked morphism is initial in L_{\mathcal{W}}((\mathbfcal{C}_{d\!% \mathrel{\text{\rotatebox[origin=c]{65}{\vrule height 2\fontdimen22\textfont2 width 0pt\rightarrow}}} })^\dagger); and (3) for every marked morphism $f:d\to b$, the induced functor

$$f^\ast:L_{\mathcal{W}}((\mathbfcal{C}_{b\! \mathrel{\text{\rotatebox[origin=c]{65}{$\vrule height 2\fontdimen22\textfont2 width 0pt\rightarrow$}}} })^\dagger)\to L_{\mathcal{W}}((\mathbfcal{C}_{d\! \mathrel{\text{\rotatebox[origin=c]{65}{$\vrule height 2\fontdimen22\textfont2 width 0pt\rightarrow$}}} })^\dagger)$$

preserves initial objects.

This is the paper's main proposed cofinality criterion, motivated by Theorem A and the decategorified criterion for marked 2-colimits. It is formulated using notions that the paper says are not yet rigorously defined in the $(\infty,2)$-categorical setting, so its proof and precise foundations remain open.