Weak higher-category structure conjecture for representable constructible polygraphs

Let $\mathbf{CPol}$ be the category of constructible polygraphs and let $\mathbf{RCPol}$ be the category of representable constructible polygraphs and strong maps. For a constructible polygraph $X$, let $X_{\mathbf O}$ denote the restriction of its presheaf to the full subcategory $\mathbf O$ of constructible atoms. A map is strong if it sends $n$-equivalences to $n$-equivalences. Weak higher-category conjecture. If $f:X\to Y$ is a strong map of representable constructible polygraphs, then $X_{\mathbf O}$ and $Y_{\mathbf O}$ admit structures of algebraic weak higher categories, and $f_{\mathbf O}:X_{\mathbf O}\to Y_{\mathbf O}$ is a functor of weak higher categories. This would provide an algebraic weak higher-categorical structure on the underlying graphs of representable constructible polygraphs; the source leaves both the precise algebraic definition and the assertion unresolved.