Conjectural factorization-algebra enhancement of the first law of black-hole thermodynamics

Let $M$ be a spacetime, let $g \in [\mathrm{Lor}(M)/\mathrm{Diff}^{+}(M)]$ be an asymptotically flat and stationary black-hole solution, and let $\mathscr{E}_g$ be the differential graded Lie algebra representing its formal neighborhood. Let $\mathrm{Obs}^{\mathrm{cl}}$ be the factorization algebra of classical observables associated to this BV classical field theory. For an asymptotically flat perturbation $h \in H^0(\mathscr{E}_g)$, black-hole first-law enhancement conjecture. The perturbation $h$ recovers a factorization-algebra enhancement of the first law of black-hole thermodynamics, with the left-hand side included from $H^0(\mathrm{Obs}^{\mathrm{cl}}(\Sigma))$ into $H^0(\mathrm{Obs}^{\mathrm{cl}}(M))$ and the right-hand side included from $H^0(\mathrm{Obs}^{\mathrm{cl}}(E_{\infty}))$. This is proposed as a factorization-algebra formulation of the first law and as a step toward a mathematical treatment of black-hole entropy; the result remains to be proved.