Briançon–Iarrobino lexsegment conjecture on maximal tangent spaces

Let $\mathbf{k}$ be a field, let $S=\mathbf{k}[x_1,\ldots,x_n]$, and for $d\in\mathbf{N}$ let $E(d)\subseteq S$ be the unique lexsegment ideal such that

$$\dim_{\mathbf{k}}\bigl(S/E(d)\bigr)=d$$

and, for some $r$, $\mathfrak{m}^{r+1}\subseteq E(d)\subseteq\mathfrak{m}^r$, where $\mathfrak{m}=(x_1,\ldots,x_n)$. For an ideal $I$, let $T(I)$ denote the tangent space at $[I]$ in the Hilbert scheme. Briançon–Iarrobino lexsegment conjecture. For every $[I]\in\mathrm{Hilb}^{d}\mathbf{A}^n$,

$$\dim_{\mathbf{k}}T(I)\leq\dim_{\mathbf{k}}T\bigl(E(d)\bigr).$$

This is the proposed extension of the maximal-tangent-space conjecture from the special lengths $d={r+n-1\choose n}$ to arbitrary $d$. The source attributes the formulation to Briançon and Iarrobino and does not report a resolution.