Herzog's canonical trace conjecture for codimension-two Cohen–Macaulay algebras

Let $S$ be either a regular local ring with residue class field $K$ or a regular positively graded $K$-algebra. Let $I$ be a perfect ideal of grade two, and let $A$ be an Hilbert–Burch matrix of $I$. Write $R=S/I$, and let $\tr(\omega_R)$ denote the trace of the canonical module of $R$. If $\mu(I)$ is the minimal number of generators of $I$ and $I_j(A)$ denotes the ideal of $j\times j$ minors of $A$, then Herzog's conjecture.

$$\tr(\omega_R)=I_{\mu(I)-2}(A)R.$$

The conjecture asserts that the canonical trace is obtained by specializing the corresponding ideal of minors from the generic maximal-minor case. The paper presents this as a conjecture posed by Jürgen Herzog; the supplied text gives no resolution status.