(All higher categories will be strict unless otherwise noted):
It is a somewhat folklorish fact that if you equip the category of $(m,n)$ categories with the cartesian monoidal structure, then a category enriched in $(m,n)$-cat is an $(m+1,n+1)$-category. More succinctly, there is an equivalence of categories
$$((m,n)\text{-Cat}, \times)\text{-Cat} \cong (m+1,n+1)\text{-Cat}.$$
There is another monoidal structure on $(m,n)\text{-Cat}$, the Crans–Gray tensor product. Denote it by $\otimes$. If I understand correctly, this should be thought of as a sort of weak version of a product, e.g. if I denote by $[1]$ the interval category, then $[1] \times [1]$ is a commutative square, whereas $[1]\otimes[1]$ is a weakly commuting square.
Given this monoidal structure, it feels natural to enrich categories in it — that is, to consider the category $((m,n)\text{-Cat},\otimes)\text{-Cat}$. For the case $m=n=2$, this turns out to recover "$3$-categories with weak interchange and everything else strict" (also called Gray categories).
Question 1: Is there somewhere in which a (Quillen) adjunction
$$((2,2)\text{-Cat}, \otimes)\text{-Cat} \rightleftarrows ((2,2)\text{-Cat}, \times)\text{-Cat} \cong (3,3)\text{-Cat}$$
is constructed? This seems (relatively) simple but I haven't been able to find a reference.
Question 2: Has anyone studied any other cases of $((m,n)\text{-Cat},\otimes)\text{-Cat}$? I am specifically interested in the case $m = \omega$ (and even more specifically, I am happy to restrict to $n=0$).
