The spinor containment conjecture for codimension-four Gorenstein schemes

Let $S$ be the polynomial ring and let $I_R$ be the ideal associated with a classifying map $\alpha\in\operatorname{SpH}_k(S)$ as in Lemma~. For a spinor set $J\cup J^c$, let $\sigma_J\in S$ be the spinor defined by

$$\bigwedge^k N_J=L\cdot\sigma_J^2,$$

where $N_J$ is the corresponding $(k+1)\times k$ submatrix and $L$ generates the cokernel of the matrix defining $\alpha$. The spinor containment conjecture. Under these assumptions, $\sigma_J\in I_R$. This is known when $R$ is reduced and when $I_R$ is generically a codimension-$4$ complete intersection, but remains unsettled in general, particularly for nonreduced or Artinian subschemes.