Formality conjecture for deformation dglas of polarizable principal bundles

Let $X_{{\bullet}}$ be a connected simplicial smooth projective variety satisfying the finite index condition. Let $P$ be a $G$-principal bundle with $\lambda$-connection, or a $G$-torsor on $|X_{{\bullet}}|$, with a fixed framing at a basepoint $x\in X_0$. Let $A^{{\bullet}}_{\eta}(X_{{\bullet}},\operatorname{ad}(P))$ be the augmented dgla controlling the deformation theory of $P$, and say that a dgla is formal in degrees at most $1\frac{1}{2}$ if it is joined to a complex with zero differential by morphisms inducing isomorphisms on $H^0$ and $H^1$ and injections on $H^2$. Formality conjecture. If $P$ is polarizable, then $A^{{\bullet}}_{\eta}(X_{{\bullet}},\operatorname{ad}(P))$ is formal in degrees at most $1\frac{1}{2}$. Furthermore, there are natural quasiisomorphisms between the Dolbeault dgla controlling deformations of the $G$-principal Higgs bundle and the de Rham and Betti dglas controlling deformations of the associated $G$-torsor. The conjecture would give a formal description of the local deformation theory and compatibility among its Dolbeault, de Rham, and Betti realizations; the supplied text gives no resolution.