Naturality conjecture for the embedding-calculus and Vassiliev isomorphism

Let $K_3$ be the space of long knots in $\mathbb{R}^3$. Let $U_{\mathcal{T}}$ and $\Sigma$ be the spaces in the preceding construction, and let

$$\iota_0:\bar H_*(U_{\mathcal{T}})\to H^*(K_3),\qquad \iota_1:H^*(K_3)\to \bar H_*(\Sigma),$$

be the maps defined there. Let $\tilde\phi:\bar H_*(U_{\mathcal{T}})\to\bar H_*(\Sigma)$ be the map obtained from the specified chain-level comparison. Naturality conjecture. The diagram with arrows $\iota_0$, $\tilde\phi$, and $\iota_1$ is commutative; equivalently, $\tilde\phi=\iota_1\circ\iota_0$. This is a conjectural compatibility between the comparison map arising from embedding calculus and the maps defining the Vassiliev-side identification; the paper relates it to a broader conjecture about embedding-calculus and additive finite-type invariants.