Minor flattening conjecture for Hilbert-scheme tangent spaces

Let $\mathbb{C}[x_1,\dots,x_d]\twoheadrightarrow R$ be a surjection to an $n$-dimensional Gorenstein algebra $R$, let $[R]$ be the corresponding point of $\operatorname{Hilb}_n\mathbb{C}^d$, and let $T$ be the structure tensor of $R$.

$$\rank\Phi_{\mathrm M,3}^C(T) - \binom{n}{3} = dn - \dim\operatorname{T}_{[R]}\operatorname{Hilb}_n\mathbb{C}^d.$$

Minor flattening conjecture. The displayed equality holds.

This conjecture proposes that the excess rank of the third minor flattening measures the excess dimension of the Hilbert-scheme tangent space at the corresponding Gorenstein algebra. The source gives computational evidence but no resolution.