The biequivalence and infinity-properad conjecture for labelled cospan categories

A labelled cospan category is a symmetric monoidal category equipped with a symmetric monoidal functor to the category of cospans satisfying the pullback condition specified in the source. A labelled infinity-cospan category is a symmetric monoidal $\infty$-category $\mathcal{C}$ equipped with a symmetric monoidal functor $\pi:\mathcal{C}\to\mathbb{C}\mathrm{sp}$ satisfying the analogous pullback condition. Let $\mathrm{Cat}_\infty^\otimes$ be the $\infty$-category of symmetric monoidal $\infty$-categories, and let $\mathcal{P}$ be the full subcategory of $(\mathrm{Cat}_\infty^\otimes)_{/\mathbb{C}\mathrm{sp}}$ on such labelled infinity-cospan categories. Labelled cospan–properad conjecture. There is a biequivalence between the $2$-category of labelled cospan categories and the $2$-category of coloured properads, and $\mathcal{P}$ is equivalent to the $\infty$-category of $\infty$-properads. The conjecture relates labelled cospan categories and labelled infinity-categories of cospans to the established notions of coloured and infinity-properads; the source suggests it as a proposed comparison, with no resolution stated.