The enriched icon construction for symmetric monoidal higher categories

Let $n$ be a positive integer and let ${\mathcal {E}}$ be a symmetric monoidal $(n+1)$-category. An ${\mathcal {E}}$-icon construction assigns to ${\mathcal {E}}$ a symmetric monoidal $(n+1)$-category of categories weakly enriched in ${\mathcal {E}}$, weak functors, and icon-like higher morphisms.

Enriched icon conjecture. For every symmetric monoidal $(n+1)$-category ${\mathcal {E}}$, there is a symmetric monoidal $(n+1)$-category ${\mathcal {E}}\text{-Icon}$ of categories weakly enriched in ${\mathcal {E}}$, weak functors, and icon-like higher morphisms.

This is proposed as part of a general construction of the correct totalities of degenerate higher categories. The case $n=1$ is supplied by the generalized icon construction in the paper, while the higher-dimensional case involves weak $n$-categories and is not yet well understood; the formalism for the required higher icon-like morphisms does not yet exist.