Exact-sequence conjecture for Spaces, the D-particle field category, and Hochschild cohomology

Let ${\bf Spaces}$ be the category of spaces, let ${\rm \ker}\,\mathbf{Q}$ be the proposed $L_\infty$ category, and let $HH({\bf Spaces})$ denote the Hochschild cohomology of ${\bf Spaces}$. Exact-sequence conjecture. There exists an exact sequence

$$1\longrightarrow {\bf Spaces}\longrightarrow {\rm \ker}\,\mathbf{Q}\longrightarrow HH({\bf Spaces})\longrightarrow 1.$$

The sequence is proposed as part of the relationship between the category of spaces, the D-particle field category, and its Hochschild cohomology; the source gives no resolution.