Radicality of determinantal ideals for unions of splitting-type strata

Let $B_1,\ldots,B_{d-1}$ be the non-trivial matrices of the direct image complexes of the versal deformation of ${\mathcal O}^{r-1}\oplus {\mathcal O}(d)$. For any collection of positive integers $r_{k_1},\ldots,r_{k_s}$, form the ideal

$$\sum_{t=1}^{s} \operatorname{minors}(r_{k_t},B_{k_t}).$$

Radicality conjecture. The ideal above is radical. This stronger assertion includes cases involving certain unions of splitting-type strata. The source records the case $r=2$ as asserted by Room in 1938 and proved in characteristic $0$ by Peskine and Szpiro.