Trace compatibility for obstruction classes of determinants

Let $A\to A_0$ be a square-zero extension with kernel $I$, and let $X\to\operatorname{Spec} A$ be a flat Artin stack such that

$$\omega_{X/A}\cong\mathscr{O}_X.$$

Write $X_0=X\otimes_A A_0$. Suppose that $P_0$ is a perfect complex on $X_0$ with determinant $L_0$. Let

$$o(P_0)\in\operatorname{Ext}^2_{X_0}(P_0,P_0\stackrel{\mathbf{L}}{\otimes} I)$$

and

$$o(L_0)\in\operatorname{Ext}^2_{X_0}(\mathscr{O},\mathscr{O}\stackrel{\mathbf{L}}{\otimes} I)$$

be the obstruction classes to deforming $P_0$ and $L_0$ to $X$. Trace compatibility conjecture. The trace map

$$\operatorname{Ext}^2_{X_0}(P_0,P_0\stackrel{\mathbf{L}}{\otimes} I)\to\operatorname{Ext}^2_{X_0}(\mathscr{O},\mathscr{O}\stackrel{\mathbf{L}}{\otimes} I)$$

sends $o(P_0)$ to $o(L_0)$. This is presented as a generally expected deformation-theoretic compatibility, and the authors indicate that they only need it for a flat family of twisted varieties; no proof or resolution is supplied here.