The Weinstein homotopy conjecture for the Liouville domains $W_{(Y,\Lambda)}$

Let $Y$ be a compact manifold without boundary, let $M\subset J^1Y$ be a closed tubular neighborhood of the zero-section, and let $\Lambda\subset(M,\ker(dz-pdq))$ be a Legendrian submanifold diffeomorphic to $Y$. For a sufficiently small closed tubular neighborhood $N_{\epsilon}(\Lambda)$, choose a contactomorphism $\phi:M\xrightarrow{\sim}N_{\epsilon}(\Lambda)\subset\operatorname{int}(M)$ satisfying $\phi^*(dz-pdq)=e^{-g}(dz-pdq)$ for some $g:M\to\mathbb{R}_{>0}$, and denote the associated Liouville domain by $W_{(Y,\Lambda)}=W_{(M,\phi)}$. Weinstein homotopy conjecture. The Liouville structure on $W_{(Y,\Lambda)}$, including the domain $W_{\operatorname{SA}}$ from the Smale example, is homotopic to a Weinstein structure. The observation is motivated by the Creation Lemma, which suggests that a single box-fold should suffice to make these domains Weinstein, but the argument is not supplied in the source; consequently, the conjecture remains open.