It’s one of the most shameful numbers in American law enforcement: According to the Department of Justice, some 400,000 “rape kits” are languishing in evidence lockers across the country because local authorities can’t afford to process them. The kits, some of them dating back to the 1980s, contain DNA evidence that could convict rapists.

Now the evidence might finally make its way to prosecutors. Last week, the White House announced a little remarked upon initiative to devote $35 million of the 2015 budget to processing unopened kits and otherwise furthering sexual assault prosecutions.

There is good evidence that pulling out old rape kits makes a difference. After New York City processed its 17,000-kit backlog in 2001, the arrest rate for rape cases jumped from 40 percent to 70 percent, reports Erin Delmore at MSNBC. In Ohio, going through 4,000 kits led to 58 cases, and in Detroit, where an 11,000-kit backlog remains, analyzing the first 10 percent of kits led law enforcement to 46 serial rapists. Since rapists have a higher recidivism rate than many other offenders, it’s especially worthwhile to reopen old case files.

This kind of progress, alas, does not come cheap. The administration’s proposed investment is only enough to make a moderate-sized dent in the issue: Testing a rape kit costs anywhere from $500 to $1,500, meaning the backlog would take hundreds of millions of dollars to clear. Under the plan announced last week, communities with a plan to start sifting through their shelves could apply for the funding, which the White House would hand out in grants.

Most states and cities don’t even know exactly how many kits they have on the shelf, or how much time and money it would take to process them. That’s not something a federal budget can address—but it is a topic of discussion in a growing number of state legislatures. Laws requiring police departments to process their untouched kits have already passed in Colorado, Illinois, and Texas, where the bill was championed by State Senator and gubernatorial candidate Wendy Davis. And at least 17 other states are considering similar measures, as Al Jazeera America reported last month.