Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set $$ K_R:= Ext_S^{s-d}(R,S), $$ where $d=\dim R$, $s=\dim S$. If $K_R$ is Cohen-Macaulay (i.e. $R$ is a canonical Cohen-Macaulay ring) and $R$ is unmixed (i.e. $dim \widehat{R}/p=d$ for all $p \in Ass \widehat{R}$), is there exist an injective homomorphism $$ R \overset{\varphi}{\longrightarrow} K_R. $$ If there exists what is $dim Coker(\varphi)$ ?

Thank you very much for your help.