Very-moving subvarieties have big classes

Let $V$ be smooth and projective, and let $W\subset V$ be a very moving subvariety: through a generic point $y\in V$, and for a generic vector subspace $W'\subset T_{V,y}$ of rank $\dim W$, there is a smooth deformation of $W$ passing through $y$ with tangent space $W'$ at $y$. Very-moving subvariety conjecture. The class $[W]$ of $W$ is big. This proposes a positivity property for very moving subvarieties, extending known results for moving curves with ample normal bundle; the supplied source gives no resolution.