Projection conjecture for large arcs and common zeros of quadrics

Let $X$ be an arc of size $2k+3$ in $\mathrm{PG}(k-1,{\mathbb F})$, and let $U$ be the subspace of quadratic forms that vanish on $X$. If

$$\dim U \geqslant {k-1 \choose 2}-1,$$

then projection conjecture. The projection of $V(U)$ to $\mathrm{PG}(3,q)$ from any $k-4$ points of $V(U)$ is contained in the intersection of two linearly independent quadratic forms. The conjecture extends the paper's observed examples, most of which project onto intersections of two quadrics; the supplied text does not indicate whether it has been resolved.