Openness conjecture for nontrivial basic de Rham classes of contact forms

Let $X$ be a closed, possibly compact, connected oriented smooth manifold, and let $v_\beta$ denote the Reeb vector field of a contact form $\beta$ on $X$. The class $[d\beta]$ belongs to the basic de Rham cohomology group $H^2_{\mathsf{basic}\,d\mathcal{R}}(X,v_\beta)$.

Openness conjecture. The set of contact forms $\beta$ on $X$ for which

$$[d\beta] \neq 0 \quad\text{in}\quad H^2_{\mathsf{basic}\,d\mathcal{R}}(X,v_\beta)$$

is open in the space of all contact forms.

The conjecture is motivated by examples on closed oriented $3$-manifolds and by the relation between nontriviality of this basic class and the linking property of the Reeb flow. Its resolution is not supplied here.