Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$ Is there a one-sided inequality?

Question

$(R,m)$ is a local ring. Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$

Answer 1

$\operatorname{inj.dim}_R R\le \operatorname{inj.dim}_R \widehat{R}$ since $R \subset \widehat{R}$ is pure.