The levelness conjecture for the power ideal algebra $R_{2,n,d}$

Let $R_{2,n,d}$ be the graded algebra under consideration, with socle ${\rm Soc}(R_{2,n,d})$ and graded component $[R_{2,n,d}]_{2d-2}$. Levelness conjecture. $R_{2,n,d}$ is a level algebra, meaning

$${\rm Soc}(R_{2,n,d})=[R_{2,n,d}]_{2d-2}.$$

The preceding computation identifies the relevant top-degree component; if the conjecture holds, the socle has dimension ${{n+d-2}\choose{n-1}}$.