Kontsevich–Soibelman cell decomposition conjecture for the Fulton–MacPherson operad

For $n\ge 2$, let $FM_2(n)$ be the Fulton–MacPherson space, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and let $P$ be the dg-operad whose standard basis is indexed by meta-trees. For a meta-tree ${\bf T}\in MT(n)$, write $\sigma_{\bf T}$ for the corresponding cell, and let $Chains_{(\sigma_{\bf T})}(FM_2)$ denote the chain subcomplex of $Chains(FM_2)$ formed by $k$-linear combinations of these cells. Kontsevich–Soibelman's cell decomposition conjecture. There is a piecewise algebraic cell decomposition of the spaces $FM_2(n)$, with cells $\sigma_{\bf T}$ labeled by $MT(n)$, such that the labeling is $S_n$-equivariant, operadic composition of cells is again a cell, there is a morphism of dg-operads

$$Chains_{(\sigma_{\bf T})}(FM_2)\to P$$

that maps every cell, up to sign, to the corresponding element of the standard basis of $P$, and for every $n\ge 2$ the decomposition is regular, meaning that the closure of every cell is homeomorphic to a closed ball. The source says that this conjecture implies Deligne's conjecture; it gives no resolution status.