When reading Lectures on Analytic Geometry, I found that in the data of an analytic ring, the underlying ring seems unnatural and the module category should be the soul, but in fact one needs the underlying ring to state the condition on the module category. After taking a look at higher algebra, it seems that I have found a way to define analytic ring without the underlying ring; namely, Propositions 7.1.2.6 and 7.1.2.7 of Higher Algebra. So I conjecture that:
For $1\le k\le\infty$, $\mathcal{A}\mapsto\mathsf{D}(\mathcal{A})$ determines an equivalence from the $\infty$-category of analytic $\mathbb{E}_k$-rings to the $\infty$-category of $\mathbb{E}_{k-1}$-monoidal $\mathsf{CondAni}$-enriched stable $\infty$-categories which are tensored over $\mathsf{CondAni}$, whose tensor product preserves colimit and the $\mathsf{CondAni}$-tensoring structure, and whose unit object $\mathbf{1}$ is a compact projective generator in the enriched sense: the enriched homomorphism $\underline{\operatorname{Hom}}(\mathbf{1},-)$ preserves sifted colimits, and every object is a colimit of some $S\otimes\mathbf{1}$'s where $S\in\mathsf{CondAni}$. In the latter category, the morphisms are defined as the $\mathsf{CondAni}$-tensored left adjoints.
If nonconnective analytic rings are reasonable, then I also conjecture a nonconnective version, removing the projectivity in the right side.
I have not yet studied higher algebra carefully, so I cannot verify these reconstruction propositions. Are they correct?
