Serre fibration conjecture for flexible Weinstein structures

Let $\eta$ be a nonempty homotopy class of nondegenerate $2$-forms on a domain or manifold of dimension $2n>4$. Let $\mathfrak{Morse}_n$ be the space of Morse functions whose critical points all have index at most $n$, and let $\mathfrak{Weinstein}_\eta^\mathrm{flex}$ be the space of flexible Weinstein structures $(\omega,X,\phi)$ whose symplectic form $\omega$ lies in $\eta$. The map

$$\mathfrak{M}:\mathfrak{Weinstein}_\eta^\mathrm{flex}\to\mathfrak{Morse}_n$$

sends a flexible Weinstein structure to its Morse function. Serre fibration conjecture. On a domain or manifold of dimension $2n>4$, the map

$$\mathfrak{M}:\mathfrak{Weinstein}_\eta^\mathrm{flex}\to\mathfrak{Morse}_n$$

is a Serre fibration with contractible fibers. The preceding theorem establishes surjectivity, path-connected fibers, and path lifting, but the stronger Serre-fibration and contractible-fiber statement is not resolved in the supplied text.