The analytic structure conjecture for the left adjoint to the underlying algebra functor

Let $\mathcal O \in \mathrm{Str}^{\mathrm{loc}}_{\mathcal T_{\mathrm{Zar}}}(\mathcal S)_{/\mathbb C}$. Let

$$\overline{\Psi} \colon \mathrm{Str}^{\mathrm{loc}}_{\mathcal T_{\mathrm{Zar}}}(\mathcal S)_{/\mathbb C} \to \mathrm{Str}^{\mathrm{loc}}_{\mathcal T_{\mathrm{an}}}(\mathcal S)_{/\mathcal H_0}$$

be the left adjoint to the underlying algebra functor. The unit $\mathcal O \to \overline{\Psi}(\mathcal O)^{\mathrm{alg}}$ exhibits $(\mathcal S,\overline{\Psi}(\mathcal O))$ as the analytification of $(\mathcal S,\mathcal O)$.

Analytic structure conjecture. For every such $\mathcal O$, the unit $\mathcal O \to \overline{\Psi}(\mathcal O)^{\mathrm{alg}}$ has this universal analytification property. This generalizes the corresponding characterization for local structures arising from points of derived schemes.

The conjecture seeks a general universal-property description of analytification in derived complex analytic geometry, beyond the geometric structures supplied by derived schemes. Its resolution is not specified in the source.