Trace equality conjecture for one-dimensional graded fiber products

Let $A$ and $B$ be the graded rings and $R=A\times_T B$ the graded fiber product considered above, with canonical modules $\omega_R$ and $\omega_{A,B}:=\omega_A\oplus\omega_B$. Assume that

$$\dim(R)=1$$

and that neither $A$ nor $B$ is regular of dimension $1$. Trace equality conjecture. Under these assumptions, the assertion of \autoref{aaa} holds, namely

$$\operatorname{tr}_R(\omega_R)=\operatorname{tr}_R(\omega_{A,B}).$$

This conjecture addresses the one-dimensional case excluded from \autoref{aaa}; the surrounding discussion notes partial results for local one-dimensional Cohen--Macaulay generically Gorenstein rings when neither ring is a discrete valuation ring. Its general status is not resolved in the supplied text.