The universal categorical spectrum conjecture for topological stabilizer codes

Let $\mathbf{TS}^*$ be a proposed categorical spectrum of topological stabilizer codes, with $\mathbf{TS}^n$ its degree-$n$ component, and let $\mathbf Q^*(\mathrm{pt})$ denote the unitary QCA spectrum at a point. Write $\mathbf{CatSp}$ for the category of categorical spectra and $\mathbf{ComAlg}$ for commutative algebra objects. Universal categorical spectrum conjecture. There is a categorical spectrum of topological stabilizer codes $\mathbf{TS}^*$ such that

$$\mathbf{TS}^{n-1}\cong\Omega\mathbf{TS}^n$$

for all $n\geq 1$, whose core is $\mathbf Q^*(\mathrm{pt})$, and such that for every other categorical spectrum $\mathbf C^*$ there is a natural-in-$\mathbf C$ isomorphism

$$\pi_0\operatorname{hom}_{\mathbf{CatSp}}(\mathbf C^*,\mathbf{TS}^*)=\operatorname{hom}_{\mathbf{ComAlg}}(\mathbf C^0,\mathbf Q^0(\mathrm{pt})).$$

This conjecture seeks a universal target category for topological stabilizer codes whose invertible part recovers the QCA spectrum. The proposed universal property would organize stabilizer codes in all dimensions and relate their bulk-boundary structure to the classification of QCAs.