Projection conjecture for point sets in projective four-space

Let $\Sigma$ be a subset of $\mathbb{P}^4$, let $n$ be such that $|\Sigma|<(n-1)^2$, and let $k$ be a positive integer. Assume that at most $k(n-1)$ points of $\Sigma$ can be contained in a curve of degree $k$. Let

$$\phi_4:\mathbb{P}^4\dasharrow\mathbb{P}^2$$

be a general projection.

Projection conjecture. At most $k(n-1)$ points of $\phi_4(\Sigma)$ can be contained in a curve of degree $k$ in $\mathbb{P}^2$.

The source states that this conjecture implies the factoriality conjecture for nodal hypersurfaces. The cited results establish factoriality under weaker bounds on the number of nodes, but the projection assertion itself is not resolved in the supplied text.