Gap-vector conjecture for Veronese embeddings

Let $X=v_d(\mathbb P^n)$ be the $d$th Veronese embedding of projective $n$-space, and define

$$\bar j:=\min\{j:g_j(X)>0\}.$$

Here $g_j(X)$ denotes the $j$th component of the gap vector of $X$, and $\operatorname{codim}X$ denotes the codimension of $X$. Gap-vector conjecture. The first nonzero component occurs at

$$\bar j=\left\lceil\binom{n+d}{d}-(n+1)+\frac{1}{2}-\sqrt{\left(n+\frac{1}{2}\right)^2+2\binom{n+2d}{2d}-2(n+1)\binom{n+d}{d}}\right\rceil,$$

and, for $\bar j\leq j\leq\operatorname{codim}X$, one has

$$g_j(X)=\binom{n+2d}{2d}-j(n+1)-\binom{\binom{n+d}{d}-j+1}{2}.$$

The conjecture asserts that the bounds previously established for the smallest index with positive gap are sharp, while also determining all subsequent components of the gap vector through codimension.