Dimension formula conjecture for ${\cal E}_{n,k}^0$

Let ${\cal E}_{n,k}^0$ denote the homogeneous degree-$k$ component of the quadratic algebra ${\cal E}_n^0$, and let $n$ and $k$ be positive integers. Dimension formula conjecture.

$$\dim {\cal E}_{n,k}^0=(2k-1)!!\binom{n}{2k}+(2k-1)!!\frac{4(k-1)}{3}\binom{n}{2k-1}+(2k-3)!!\frac{(k-1)(k-2)(16k-3)}{9}\binom{n}{2k-2}+\cdots.$$

The displayed expression records the leading terms of a proposed dimension formula; the ellipsis leaves further terms unspecified. The source gives no resolution.