Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true?

I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein.

Question 2. What about replacing "$H^i_m(R)=E(k)$" by "$H^i_m(R)$ is injective"?

Answer 1

Q1, Q2: Yes, this works even more generally:

If $(R,m)$ is local and there is some integer $n$ such that $H^i_m(R) =0$ for $i \neq n$ and $H^n_m(R)$ has finite injective dimension, then $R$ is Gorenstein.

This follows from Theorem 2.5 of http://arxiv.org/pdf/1204.2394v5.pdf by taking $M=R$ and $a=m$.