Scheme-theoretic multiplicity formula for pullbacks of higher associated varieties

Let $\lambda=[\lambda_1,\ldots,\lambda_n]$ be a partition, let $m_1(\lambda)$ denote the multiplicity of its parts equal to $1$, and let $\mathcal{F}_j(\lambda)$ be the family of descendants defined in the paper. For $\lambda'=[\lambda'_1,\ldots,\lambda'_n]\in\mathcal{F}_j(\lambda)$, let $m(\lambda',\lambda)$ be the multiplicity defined by

$$m(\lambda',\lambda)=\#\{(\iota_1,\ldots,\iota_n):\iota_i\in\{0,1\},\ \iota_1+\cdots+\iota_n=j,\ [\lambda'_1+\iota_1,\ldots,\lambda'_n+\iota_n]=[\lambda_1,\ldots,\lambda_n]\}.$$

Here $\varPsi_{d,r}$ is the apolar map, $\mathrm{CH}_j(\Delta_\lambda)$ is the relevant higher associated variety, and $(\Delta_{(\lambda'_1+1,\ldots,\lambda'_n+1)})^{\vee}$ denotes the dual variety. Scheme-theoretic multiplicity formula. If $j\leq n-m_1(\lambda)$, then

$$\overline{\varPsi_{d,r}^{-1}(\mathrm{CH}_j(\Delta_{\lambda}))}=\bigcup_{\lambda'\in\mathcal{F}_j(\lambda)}m(\lambda',\lambda)\,\overline{\varPsi_{d,r-j}^{-1}(\mathrm{CH}_0(\Delta_{\lambda'}))}=\bigcup_{\lambda'\in\mathcal{F}_j(\lambda)}m(\lambda',\lambda)\,(\Delta_{(\lambda'_1+1,\ldots,\lambda'_n+1)})^{\vee}$$

with equality holding scheme-theoretically. The formula gives a geometric description of the components and their multiplicities in these pullbacks; it is proposed on the basis of experimental verifications and general considerations about singular loci, and its resolution status is not established in the supplied text.