Let $R$ be a local Cohen-Macaulay domain of dimension at least $2$. Let $M$ be a maximal Cohen-Macaulay $R$-module such that localization of $M$ at every height $1$ prime ideal of $R$ is free. Consider a short exact sequence $0\to F \to M \to I \to 0$ where $F$ is a free $R$-module and $I$ is a non-zero ideal of $R$ such that localisation of $I$ at every height $1$ prime ideal of $R$ is free. Then, is $I$ a maximal Cohen-Macaulay $R$-module?
By depth along short exact sequence (https://stacks.math.columbia.edu/tag/00LX), clearly $\text{depth } I \ge \dim R -1$, but I cannot decide if $\text{depth } I = \dim R$ or not.
Please help.