Maximal loss of quadrics under projection of Veronese varieties

Let $X= u_d(\mathbb{P}^n)$ be a Veronese embedding, with $n>2$, and consider successive projections away from points of $X$. At each step, the number of quadrics in the vanishing ideal that are lost is bounded by the codimension of the projected variety.

Maximal-loss conjecture. In each projection step, one loses the maximal number of quadrics, namely the codimension of the projected variety. Equivalently, successive general-point projections lose the codimension number of quadrics at every step. This conjecture holds for $n=2$.

If true, the conjecture gives a direct computation of the quadratic persistence of Veronese embeddings. The paper states that the problem is open for $n>2$ and relates it to the dimensions of certain catalecticant varieties.