I am not sure what "relative to the homicide rate" is supposed to prove. The homicide rate is not the same as the homicide conviction rate, or put another way, perhaps we have a lower incarceration rate relative to our homicide rate compared to other countries because we are solving a smaller fraction of homicides compared to those countries.

(Also possible, but I suspect less likely, is that our homicide rate is higher but not in proportion to the number of murderers; maybe we have a bigger problem with serial killers than other countries, and thus have fewer murderers to incarcerate relative to the homicide rate.)