The oriented connected-endomorphism conjecture for the 2-sphere

Let $\mathbb{L}_{SO}^\otimes$ be the proposed oriented $\infty$-operad, and let $\Omega^2\operatorname{Bord}_3^{SO}$ be the symmetric monoidal $\infty$-category of oriented 3-dimensional bordisms. Write $\operatorname{End}_{\Omega^2\operatorname{Bord}_3^{SO}}^\otimes(S^2)$ for the oriented $\infty$-endomorphism operad of $S^2$, and $\operatorname{End}_{\Omega^2\operatorname{Bord}_3^{SO}}^{\otimes,\operatorname{conn}}(S^2)$ for its suboperad consisting of connected bordisms. The oriented connected-endomorphism conjecture. There is a map of $\infty$-operads

$$f_{SO}:\mathbb{L}_{SO}^\otimes\to\operatorname{End}_{\Omega^2\operatorname{Bord}_3^{SO}}^\otimes(S^2),$$

whose image is $\operatorname{End}_{\Omega^2\operatorname{Bord}_3^{SO}}^{\otimes,\operatorname{conn}}(S^2)$, and $f_{SO}$ is an equivalence

$$\mathbb{L}_{SO}^\otimes\overset{\sim}{\to}\operatorname{End}_{\Omega^2\operatorname{Bord}_3^{SO}}^{\otimes,\operatorname{conn}}(S^2).$$

This would identify the proposed oriented operad with the connected part of the full oriented $\infty$-endomorphism operad of the 2-sphere. The source gives no resolution evidence, so the conjecture remains open.