Formality criterion for simply connected finite CW complexes

Let $X$ be a simply connected finite CW complex, and let $X_B$ denote the constant $\infty$-stack whose values are the Poincaré $\infty$-groupoid of $X$. Formality criterion conjecture. The following properties are equivalent: (1) for all $n$ and every $1$-connected namhs ${\cal V}$, the pre-weight-filtered $n$-stack

$$\underline{\operatorname{Hom}}(X_B,(V_{B,\mathbf C},W))$$

satisfies condition ${\bf A1}$; and (2) the differential graded algebra associated to $X$ by the theory of Sullivan and Morgan is formal. The paper says that vague calculations support this conjecture and relates it to the formality theorem for smooth projective varieties.