Theory of weak enrichment in higher categories Has there been work towards a general theory of weak enrichment in higher categories? To be more pointed, has there been any work towards trying to make sense of statements such as

There is a (weak) $(n+1,r+1)$-category $\mathcal V\mathbf{Cat}$ of categories enriched in a monoidal $(n,r)$-category $\mathcal V$.

generalising the fact that there is a $2$-category of categories enriched in an ordinary monoidal category, and how I imagine there is a (weak?) $3$-category of categories enriched in a monoidal bicategory.
I know there's work by Gepner and Haugseng (and more in this direction) which makes sense of an $(\infty,1)$-category of categories enriched in a monoidal $(\infty,1)$-category, though from my (limited) understanding there is little care here for the non-invertible "enriched natural transformations" which would exist in the corresponding $(\infty,2)$-category of enriched categories, and a lot of the work is put into ensuring that the equivalences of enriched categories correspond to the correct notion of fully faithful and essentially surjective functors.
Besides these, and other texts that address special cases (e.g., using a notion of weak enrichment to create a theory of weak $n$-categories), I haven't found anything that addresses a more general notion of enrichment in higher monoidal categories. Perhaps I just don't know how to look for them, but it also leads me to wonder: is there reason to study weak enrichment in this way? Or conversely, is there no good reason to be concerned with this? (e.g., problems where this would come in handy don't come up often / don't exist, or the $(\infty,1)$-category theory of enrichment is sufficient for most practical purposes, etc.)
Edit: Harry Gindi pointed out that Gepner and Haugseng do provide some work in the direction of higher categorical structure. In particular (this is their example 7.4.11), if $\mathcal V$ is a $\Bbb E_2$-monoidal (presentable) $(\infty,1)$-category, then the $(\infty,1)$-category $\mathcal V\mathbf{Cat}$ of $\mathcal V$-enriched categories will be monoidal also; moreover, if $\mathcal V$ is closed, then so is $\mathcal V\mathbf{Cat}$. Especially, the latter is self-enriched to provide a "$\mathcal V$-$(\infty,2)$-category" of $\mathcal V$-enriched categories (in that between $\mathcal V$-enriched categories is a $\mathcal V$-enriched category of functors).
While this provides $\mathcal V\mathbf{Cat}$ with a nice self-enrichment in instances where the enriching category is nice, this is somehow giving more than I asked for given less than I have. For instance, if $\mathcal V$ is just an ordinary monoidal $1$-category, then $\mathcal V\mathbf{Cat}$ will always be a $2$-category, though with additional assumptions on $\mathcal V$ (take it to be a Bénabou cosmos, for instance) it can also be given an internal hom (making it a "$\mathcal V$-enriched $2$-category").
If $\mathcal V$ is a general monoidal $(\infty,1)$-category, is it possible that $\mathcal V\mathbf{Cat}$ will to naturally an $(\infty,2)$-category?
Edit 2 (for clarification): While Harry's answer does a great job of elaborating on the work of Gepner and Haugseng, it doesn't really answer my question (as far as I know). I'm not trying to inductively define $(n+1,r+1)$-categories through iterated weak enrichment; rather, I'm just trying to see if there are constructions which, given for example a (nice?) monoidal $n$-category $\mathcal V$, produces the $(n+1)$-category $\mathcal V\mathbf{Cat}$ of categories enriched in $\mathcal V$. I mentioned Gepner and Haugseng's work because it's a step in this direction, but I don't think they were trying to address this generality.

Since my post is getting a bit long, I'll keep a summary of the questions I'm really trying to ask:

*

*Is there existing work towards a general theory which describes the $(n+1,r+1)$-category of categories enriched in a monoidal $(n,r)$-category?

*If not, is this because there is no merit to this endeavour in the first place, or is it because this is just a difficult thing to accomplish in general?

 A: If you take a look  at Gepner-Haugseng Corollary 5.7.12, the first thing you'll notice is that the functor $\mathbf{Cat}^{(-)}_\infty:\mathbf{MON}^{\operatorname{lax}}_∞\to \mathbf{CAT}_\infty$ is lax monoidal and sends $\mathcal{O}$-algebras to $\mathcal{O}$-monoidal categories for any symmetric operad $\mathcal{O}$.  Another way to say this is that enrichment takes an $\mathcal{O}\otimes E_1$-monoidal category to an $\mathcal{O}$-monoidal category.
Now here's a neat thing to note: If we restrict this to the subcategory of presentably monoidal $\infty$-categories, we get the stronger statement that the functor $\mathbf{Cat}^{(-)}_\infty$ carries $\mathbf{Mon}^{\operatorname{pr},\operatorname{lax}}_\infty$ to $\mathbf{Pr}^L_\infty$ and is lax monoidal, sending the tensor product of presentably monoidal $\infty$-categories to the tensor product of presentable $\infty$-categories.
In particular, we now get the same statement as before, namely that for any symmetric operad $\mathcal{O}$, the functor $\mathbf{Cat}^{(-)}_\infty$ carries presentably $\mathcal{O}\otimes E_1$ monoidal categories to presentably $\mathcal{O}$-monoidal categories. Ok, so how does this answer your question?
Well, presentably $\mathcal{O}$-monoidal categories are closed, namely for each arity, we have an n-fold tensor product $\otimes^n:C^{\otimes n}\to C,$ and if we choose a family of $n-1$ objects $(c_1,\dots,c_{\hat{i}},\dots,c_n)$ (omitting the ith index), we obtain a colimit-preserving functor $C\to C$, which now admits an adjoint, the internal hom (if you aren't symmetric monoidal, these can all vary in complicated ways).
Now applying this to $\mathbf{Cat}^{\mathcal{V}}_\infty$, for $\mathcal{V}$ presentably $E_n$-monoidal, we see that it is canonically enriched over itself.  Moreover, the assignment $\ast \mapsto \mathbb{1}_{\mathcal{V}}$ extends to a lax monoidal functor $\mathcal{S}\to \mathcal{V}$, which we can use to understand the 'underlying $(\infty,2)$-category $\widetilde{\mathbf{Cat}}^{\mathcal{V}}_\infty$'.
You can then iterate this procedure to produce an $E_{n-k}$ presentably-monoidal $(\infty,k)$-category of $k$-fold iterated enriched categories.
So I'm a bit confused what you're talking about.  You do indeed get all of the correct kinds of natural transformations.  There is a problem in the case where you choose $\mathcal{V}$ to be $\mathbf{Cat}_{\infty,n}$, but this is not an enriched story.
It is expected (or already proven, depending on who you ask) that there is another special biclosed $E_1$ presentably-monoidal structure on $\mathbf{Cat}_{\infty,n}$ for each $n\leq \omega$.  This is called the Gray tensor product, or the lax Gray tensor product.  This is not an enriched tensor product at all.  It can't commute with or distribute over the Cartesian product.  It is sui generis, and its right adjoints classify functors with lax or oplax natural transformations between them.  This is an extremely important construction, but it has very little to do with enrichment.  It looks like the easiest way to construct it is actually by first inducing it on $\mathbf{Cat}_{\infty,\omega}$, then inducing it on each finite $n$ by localization.
