Embedded and unrestricted Legendrian stop moduli conjecture

Let $Y$ be the ambient space and let $\Lambda_n$ be the specified Legendrian stop. Write $\mathcal L^{\mathrm{emb}}_{\Lambda_n}(Y)$ for the moduli space of stops with embedded Lagrangian projection and $\mathcal L_{\Lambda_n}(Y)$ for the unrestricted stop moduli space. Embedded-stop equivalence conjecture. The spaces are homotopy equivalent:

$$\mathcal L^{\mathrm{emb}}_{\Lambda_n}(Y)\simeq \mathcal L_{\Lambda_n}(Y).$$

The conjecture would show that restricting to stops with embedded Lagrangian projection does not change the homotopy type, but the source gives no proof or resolution.