I took apart the DP-10 and looked at the MIDI values generated by the potentiometer when I manually rotated it through its full range. The values ranged from 4 to 127, which was obviously better than the what I was getting previously, i.e., 23 to 127.
This suggested that when the pedal is not depressed something is preventing it (and hence the potentiometer) from returning to the rest position at which its minimum MIDI value is output. It didn’t take me long to notice that the felt pad against which the pedal rests was too thick. This felt pad is to prevent the pedal from contacting the pedal case when fully the pedal returns to the rest position.
The solution then was simple — I used the extended blade of a box cutter to slowly shave layers off the felt pad until it was thinner by about 50%. Goodbye MIDI-OX and MIDI Yoke!