Monoidality conjecture for the shape functor of constructible polygraphs

Let $\mathbf{CPol}$ be the category of constructible polygraphs, with monoidal structures given by the lax Gray product $\scaleobj{0.75}{\boxtimes}$ and the join $\star$, and let $\omega\mathbf{Cat}$ be the category of $\omega$-categories with the corresponding monoidal structures. The shape functor is denoted by $-_{\omega}:\mathbf{CPol}\to\omega\mathbf{Cat}$. Monoidality conjecture. The functor $-_{\omega}$ is monoidal both from $(\mathbf{CPol},\,\scaleobj{0.75}{\boxtimes},1)$ to $(\omega\mathbf{Cat},\,\scaleobj{0.75}{\boxtimes},1)$ and from $(\mathbf{CPol},\,\star,\emptyset)$ to $(\omega\mathbf{Cat},\,\star,\emptyset)$. This would show that the combinatorial shape construction preserves both fundamental monoidal structures relevant to higher categories; the source presents it as an expected result, with no proof or resolution supplied.