Characterization of generically reduced elementary components for two-step nestings

Fix an integer $n\geqslant 4$ and consider a nesting

$$\underline{\boldsymbol{h}}=((1,s),(1,k,2)).$$

Let $V_{\underline{\boldsymbol{h}}}=\overline{H^n_{\underline{\boldsymbol{h}}}}\times \mathbb{A}^n$, where $H^n_{\underline{\boldsymbol{h}}}$ is the corresponding nested Hilbert-function stratum. Also write $\boldsymbol{h}^{(2)}$ for the second Hilbert function in the nesting.

The nesting conjecture. The variety $V_{\underline{\boldsymbol{h}}}$ is a generically reduced elementary component if and only if

$$4\leqslant k\leqslant n,\qquad 2\leqslant s\leqslant k-2.$$

On the other hand, if $\boldsymbol{h}^{(2)}(2)=1$, then $V_{\underline{\boldsymbol{h}}}$ is contained in the smoothable component.

The claim records the expected component structure for these nested Hilbert schemes. The supplied text gives no resolution status or evidence beyond the claim itself, so it is treated as open.