My grandson dropped something in the toilet which plugged it up. I couldn't fix it with a plunger(perhaps because I lacked confidence and did it in a wimpy manner with a cheap plunger) but the apartment maintenance staff were able to unplug it with a plunger. Instead of pushing it further down the pipe, the plunger brought it up and out.
As I write this I'm wondering if I remembered correctly. I didn't see them doing it. You might want to ask for advice from a plumber. Calling doesn't mean you have to have them come out. My father was an appliance repairman and he often gave advice to callers so that they could do it themselves. It was good PR.
After I wrote this I thought the problem through. I also was thinking that plunging should push it down further. However, when I've plunged for other reasons such as too much paper or too large of a bowel movement, everything comes back up before it goes back down. To remove the fingernail scrubber you'd probably have to grab in quickly.