Briançon–Iarrobino conjecture on maximal tangent spaces of punctual Hilbert schemes

Let $N\geq 1$ and let $\mathfrak{m}=(x_1,\ldots,x_N)\subset\mathbb{C}[x_1,\ldots,x_N]$. Briançon–Iarrobino conjecture. The ideal $\mathfrak{m}^k$ has the maximum dimension tangent space among all elements of

$$\operatorname{Hilb}^{\binom{N-1+k}{N}}(\mathbb{A}^N).$$

This is described as a well-known long-standing conjecture, attributed in the source to Briançon and Iarrobino (1978); the source does not state a resolution.