Compatibility conjecture for the critical and framed-sheaf obstruction theories

Set $\boldsymbol{k}={\mathbb C}$. Let

$$\operatorname{Quot}_{{\mathbb A}^3}(\mathscr O^{\oplus r},n)=\{\mathrm{d}f=0\}\subset U=U_{r,n,3}$$

be the critical-locus presentation, with critical obstruction theory

$${\mathbb E}_{\operatorname{crit}}=\bigl[T_U\big|_Q\xrightarrow{\operatorname{Hess}(f)}\Omega_U\big|_Q\bigr]\to {\mathbb L}_Q.$$

Let ${\mathbb E}$ be Oprea's symmetric perfect obstruction theory on $\mathrm{Fr}_{r,n}({\mathbb P}^3)$, and let $\eta$ be the isomorphism of Theorem identifying the relevant framed-sheaf moduli space with $Q$.

The compatibility conjecture. The isomorphism $\eta$ induces an isomorphism of perfect obstruction theories

$${\mathbb E}_{\operatorname{crit}}\cong\eta^*{\mathbb E}$$

over the truncated cotangent complex of $\operatorname{Quot}_{{\mathbb A}^3}(\mathscr O^{\oplus r},n)$.

This conjecture asserts compatibility between the canonical critical-locus obstruction theory on the Quot scheme and the symmetric obstruction theory arising from framed sheaves. It is proposed as a higher-rank version of the conjecture cited in the source, and no resolution is given here.