If this were true, then CCW should still be asssociated with lower levels of crime. While this relationship shows up in the 1977-1992 data, it disappears when you add 1993-1999 data.
However, with the 1993-99 trends, crime was making significant drops with or without CCW due to economic and demographic factors. If CCW caused a five percent drop in crime during, say, 1987 to 1993, but crime dropped sixty percent from 1993 to 1999 because of national trends, you can lump the two together to trivialize the drop from CCW.