Conjecture on higher vanishing of A-infinity operations for manifolds

Let $M$ be an orientable $N$-dimensional $k$-connected compact manifold, with $\dim H^{k+1}(M)=1$. Let $l\geq 3$ and assume

$$N\leq (l+1)k+4.$$

An $A_\infty$-minimal model of $\Omega^*(M)$ is a minimal $A_\infty$-algebra model whose higher operations are denoted by $m_p$. The manifold vanishing conjecture. Under these assumptions, $\Omega^*(M)$ has an $A_\infty$-minimal model with

$$m_p=0\quad\text{for }p\geq j.$$

The statement appears in the paper's section on conjectures and is motivated by an extension of Cavalcanti's formality result, but the source does not define $j$ or state whether the conjecture has been resolved.