Can I ask for a trial on a dead docket case?

Yes, but why would you want to?

When a case is placed on the dead docket, the State rarely intends to bring it to trial. It's a way for prosecutors to get rid of cases they don't want to try, but also don't want to dismiss.

It may be that you want a trial to prove that you're not guilty. Good luck--there's always a risk that the jury will convict you when you go to trial. If that risk is acceptable and it's important for you to beat the charges publicly, then you can revive your case and demand that it go to trial.

If you don't really want a trial but the open case is preventing you from getting hired, then you have two options. First, you can ask the State to nolle pros the case. There's a time period in which they can refile a case form a nolle pros, but the case will show as closed on a background check.

Second, after your case has been on the dead docket for 12 months, you can apply for a record restriction and the case will come off your record, even though it's technically still open.