Tangency flattening conjecture for Hilbert-scheme tangent spaces

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

$$\rank\tang^C(T) - n(n-1)(n-2) = dn - \dim\operatorname{T}_{[R]}\operatorname{Hilb}_n\mathbb{C}^d.$$

Tangency flattening conjecture. The displayed equality holds.

This conjecture would identify the excess rank of the tangency flattening with the excess codimension of the tangent space to the Hilbert scheme. The source presents it as a future direction; no resolution is supplied.