Hilbert-coefficient conjecture for powers of the square of the homogeneous maximal ideal

Let $S_n=k[x_1,\ldots,x_n]$ and let $I=(x_1,\ldots,x_n)^2$. The Hilbert coefficients $e_i(S_n/I^s)$ are defined from the eventual polynomial expression for $l(S_n/I^s)$ in the binomial basis.

Square-of-the-maximal-ideal conjecture. The coefficients satisfy:

$$e_i(S_n/I^s)\ne0\quad\text{exactly if}\quad i\le\left\lfloor\frac n2\right\rfloor.$$

If $n\ge2j$, then

$$e_j={n-j\choose j}2^{n-2j}.$$

Consequently,

$${2s-1+n\choose n}=\sum_{j=0}^{\lfloor n/2\rfloor}(-1)^j2^{n-2j}{n-j\choose j}{s+n-j\choose n-j}.$$

The authors report having checked the conjecture for $n\le11$ and identify combinatorial proofs as desirable; the general statement remains open in the source.