Bucket, towels, pliers, screwdriver, pipe cleaner or long brush, wet/dry vacuum.
Many people think eggshells "sharpen" disposal blades. In a dishwasher, they are a nightmare. The membrane inside the shell gets stringy and creates a net that traps other debris. dishwasher pipe clogged
If that works, you just saved yourself an hour of labor. Bucket, towels, pliers, screwdriver, pipe cleaner or long