Generic absence of sky catastrophes for homotopies of Reeb vector fields

Let $M$ be a compact contact manifold, and let $\{X_t\}$, $t\in[0,1]$, be a smooth homotopy through Reeb vector fields. A sky catastrophe is the phenomenon excluded by the stated genericity condition for the family. Generic sky-catastrophe conjecture. There is a $C^0$-nearby smooth family $\{X'_t\}$, $t\in[0,1]$, satisfying $X'_i=X_i$ for $i=0,1$, such that $\{X'_t\}$ has no sky catastrophes. This would rule out Fuller’s sky catastrophe after an arbitrarily small perturbation relative to the endpoints; the supplied text does not state whether the claim is known or unresolved.