The FBI data doesn't back it up. Not when looking at murder, or assault, or armed robbery.
BTW, I couldn't find that text in the link I provided. Are you looking at a different study?
Oh wait, I found it:
http://www.ncjrs.gov/pdffiles1/nij/grants/204431.pdf An unpublished paper by a UoP prof.
"Because the ban has not yet reduced the use of LCMs in crime, we cannot clearly credit the ban with any of the nation’s recent drop in gun violence." -- check. Banning LCMs doesn't reduce crime

"Should it be renewed, the ban’s effects on gun violence are likely to be small at best and perhaps too small for reliable measurement. AWs were rarely used in gun crimes even before the ban. LCMs are involved in a more substantial share of gun crimes, but it is not clear how often the outcomes of gun attacks depend on the ability of offenders to fire more than ten shots (the current magazine capacity limit) without reloading." -- check, check. Substitution, and second gun offsets this ban.
"Approximately 40 percent of the semiautomatic handgun models and a majority of the semiautomatic rifle models being manufactured and advertised prior to the ban were sold with LCMs or had a variation that was sold with an LCM (calculated from Murtz et al., 1994)" -- I would think this number has reached >50%, especially within the last 6 months. Any ban on LCMs would come up against the 'in common, lawful use' test set forth in Heller. Check, check, check.
I love this bit-
"Pistol Grip
Allows the weapon to be “spray fired” from the hip. Also helps stabilize the weapon during rapid fire."
Put your hand around an empty can of soda. Which is more comfortable while holding it at your hip, with your fingers down and the can horizontal, or at an awkward angle with the can pointing up?
"Looking at the nation’s gun crime problem more broadly, however, AWs and LCMs were used in only a minority of gun crimes prior to the 1994 federal ban, and AWs were used in a particularly small percentage of gun crimes." -- check. Solution looking for a problem.