Deformation-theoretic conjectures for geometric n-stacks

Let $A$ be a geometric $n$-stack, and let $D=\operatorname{Spec}(k[\epsilon]/(\epsilon^2))$. Let $TDef_A$ denote the $(n+1)$-category of deformations of $A$ parametrized by $D$. Let $TA$ be the tangent spectrum of $A$, let $\Omega^{-1}TA$ be its $1$-fold delooping, and let $\Gamma(A,-)$ denote sections over $A$. Deformation-theoretic conjecture. (1) The $(n+1)$-groupoid $TDef_A$ has a structure of a spectrum; that is, it is the $N$-fold looping of an $(N+n+1)$-groupoid. (2) If $A$ is a smooth geometric $n$-stack, there is a natural equivalence of $(n+1)$-categories

$$TDef_A\cong\Gamma(A,\Omega^{-1}TA).$$

For a smooth scheme, this reduces to $TDef_A\cong H^1(A,TA)$. (3) In general, there is a cotangent complex $Cot(A)$ that is a spectrum over $A$, and

$$TDef_A\cong\Gamma(A,\Omega^{-1}Cot(A)).$$

These statements aim to provide the expected deformation theory for geometric stacks, extending the classical cotangent complex for schemes and the cotangent complex of Laumon–Moret-Bailly for $n=1$. The source gives no evidence of resolution.