Let $M$ be a closed manifold admitting an almost complex structure and let $\mathcal{S}$ be the set of smooth structures on $M$ with the $C^{\infty}$ topology.
When constructing a moduli space of pseudoholomorphic curves on $M$, the almost complex structure on $M$ is allowed to vary in the space $\mathcal{J}_{\tau}(M,\omega)$ of $\omega$-tame almost complex structures, for some symplectic form on $M$. This forms an open subset of $\mathcal{S}$.
McDuff and Salamon (section 3.1) say that in fact any open (in the $C^{\infty}$ topology) subset $\mathcal{J}\subset \mathcal{S}$ of almost complex structures works just as well as $\mathcal{J}_{\tau}(M,\omega)$. In particular, a symplectic form on $M$ is not necessary.
My question is about replacing $\mathcal{J}_{\tau}(M,\omega)$ with a larger set when constructing a moduli space.
Question: Let $\mathcal{J}(c_1)\subset \mathcal{S}$ be the subset consisting of almost complex structures on $M$ with first Chern class $c_1$. Is $\mathcal{J}(c_1)$ an open subset of $\mathcal{S}$?
Note: Since $\mathcal{J}_{\tau}(M,\omega)$ is contractible, the first Chern class $c_1$ is constant in $\mathcal{J}_{\tau}(M,\omega)$, as shown here. Therefore, for any symplectic form $\omega$ on $M$ (if one exists), $\mathcal{J}_{\tau}(M,\omega)$ will always be contained in $\mathcal{J}(c_1)$ for some $c_1$.