Kudos - this worked for me too!
I took a cotton swab/Q-Tip and dipped it into a little ISO-propyl rubbing alcohol and gently inserted the damp tip into each of the headphone jacks. For each jack, I inserted and removed the damp swab a few times inserting only till it bumped against something inside.
BTW I googled casio px-860 no audio and this thread came up as first link.
Many thanks Pianosaur!