Comparison of simplicial and E-infinity formal-group Hochschild homology

Let $\formalgroup$ be a formal group over a perfect field $k$. For a simplicial commutative $k$-algebra $A$, let $\theta(A)$ be its underlying $E_\infty$-algebra, and write $\operatorname{HH}_{E_\infty}^{\formalgroup}(\theta(A))$ for the $E_\infty$-$\formalgroup$ Hochschild homology defined using spectral mapping stacks. Formal-group Hochschild comparison conjecture. There is a natural equivalence

$$\theta(\operatorname{HH}^{\formalgroup}(A)) \to \operatorname{HH}_{E_\infty}^{\formalgroup}(\theta(A)).$$

Equivalently, the underlying $E_\infty$-algebra of the $\formalgroup$-Hochschild homology agrees with the $E_\infty$-$\formalgroup$-Hochschild homology of $A$ viewed as an $E_\infty$-algebra. This conjecture is the precise comparison between the two constructions introduced in the paper, and the supplied text does not report a resolution.