Homotopic Hopf–Galois extension Quillen equivalence conjecture

Let $H$ be a bimonoid in a monoidal model category $\mathbf M$. Suppose that the category $\mathbf{Alg}_{H}$ of $H$-comodule algebras admits a model structure for which the coinvariants functor

$$\operatorname{Coinv}:\mathbf{Alg}_{H}\to\mathbf{Alg}$$

is right Quillen. Let $A$ be an $H$-comodule algebra, and let $\varphi:\operatorname{Triv}(B)\to A$ be a morphism in $\mathbf{Alg}_{H}$, with associated Galois map

$$\beta_{\varphi}:A\underset B\otimes A\to A\otimes H.$$

If $\varphi$ is a homotopic $H$-Hopf-Galois extension, then the homotopic Hopf–Galois extension conjecture. the induced pair

$$(\beta_{\varphi})_*:\mathbf D(\varphi)\to \mathbf M_A^{W_{\rho}}:\ -\underset{A\otimes H}{\square}(\beta_{\varphi})_*(A\underset B\otimes A)$$

of functors is a Quillen equivalence. This is proposed as a homotopical analogue of the classical Hopf–Galois result attributed to Schauenburg; the source does not specify the additional strong conditions on $\varphi$ under which the claim is expected to hold, nor does it provide evidence that the conjecture has been resolved.