Cisinski-style conjecture on cellularly generated localisers being sketchable

Let $C$ be an $\infty$-category, and let $W\subseteq (\operatorname{Cat}\!\slice C)^{\Delta^1}$ be a set of maps satisfying three conditions: it is stable under 2-out-of-3 and retracts; it contains every map $[\omega:\point\rightarrow I\stackrel{U}{\rightarrow} C]$ when $I$ has final object $\omega$; and whenever $I\stackrel{h}{\rightarrow} J\rightarrow K\rightarrow C$ has $[h\slice k:I\slice k\rightarrow J\slice k\rightarrow C]\in W$ for every $k:K$, one has $h\in W$. Suppose moreover that $W$ is of cellular generation, meaning that it is minimal among sets of maps of diagrams containing a small subset of $W\cap(\operatorname{Cat}\!\coneslice C)$ and satisfying these three properties. Cisinski-style sketchability conjecture. Then $W$ is the set of weak equivalences for a sketch on $C$. This proposes a converse to the sketchability criterion, extending Cisinski's work on fundamental localisers; its resolution is not supplied here.