The iterated icon construction for slightly strict higher categories

Fix $n \geq 1$. Let ${\mathcal {E}}_{n,0}$ denote the symmetric monoidal $(n+1)$-category of $n$-categories. For each $k \geq 0$, define ${\mathcal {E}}_{n,k}$ recursively by applying the icon construction to the preceding symmetric monoidal $(n+1)$-category.

Iterated icon conjecture. We define a symmetric monoidal $(n+1)$-category ${\mathcal {E}}_{n,k}$ for each $k \geq 0$ by

$${\mathcal {E}}_{n,0}=n\text{-Cat},\qquad {\mathcal {E}}_{n,k}=({\mathcal {E}}_{n,k-1})\text{-Icon}.$$

The objects of ${\mathcal {E}}_{n,k}$ are intended to model slightly strict $(n+k)$-categories. This construction is proposed because iterated icons should give the appropriate lower-dimensional totalities of degenerate higher categories, although the weak higher-category formalism needed when $n>1$ is not yet sufficiently developed.