Multiplicativity conjecture for discriminant algebras and norm functors

Let $\Delta$ be the discriminant algebra operation, sending rank-$n$ algebras $\mathcal{A}\to\mathcal{B}$ to quadratic algebras $\mathcal{A}\to\Delta_{\mathcal{B}/\mathcal{A}}$. Let $\mathcal{A}\to\mathcal{B}$ be a rank-$n$ algebra and $\mathcal{B}\to\mathcal{C}$ a rank-$m$ algebra, so that $\mathcal{A}\to\mathcal{C}$ has rank $mn$. Discriminant–norm multiplicativity conjecture. There is an isomorphism of quadratic $\mathcal{A}$-algebras

$$\Delta_{\mathcal{C}/\mathcal{A}}\cong \mathrm{Nm}_{\mathcal{B}/\mathcal{A}}(\Delta_{\mathcal{C}/\mathcal{B}})\ast\Delta_{\mathcal{B}/\mathcal{A}}^{\ast m}.$$

This conjecture proposes a compatibility between the discriminant algebra operation and the norm functor under composition of finite-rank algebra extensions; its status is not resolved in the supplied context.