A possible solution would be using an plastic sheet or standard DVD/CD case as an intermediate protection layer, but I do not know if this would affect the results as shown by Parkhomov and coworkers. There does not seem to be much detailed accessible information on the methodology; from my past attempts, both making sure that the disks do not get scratched and keeping track of existing light scratches proved challenging.
For most other researchers who have used CR-39 sheets to observe loosely similar particle tracks&spots, there is usually a need for the sheets to be very close to the cathode during active conditions, and to etch them in a temperature-controlled hot caustic solution.
EDIT: gif animation from the previous video, unrelated to the comment, showing how the reactive spiky surface appeared to burn/combust in air. I tried looking at it closer once and it had a dark green color (presumably from NiO). In later attempts I used Ni25Cu75 alloy and the color was different (blue-gray, similar to how coins made of the same material look when oxidized).