If I knew it wouldn't hurt the axe or circuitry I'd spray with contact cleaner? That used to/still does work on regular switches! I'm new to these axes so don't do it till an expert replies!
I suggest you open a support ticket to work directly with Line 6 on this. Under the Support tab at the top of this page, select Contact Technical Support. Then make the same selection on the subsequent page.
There were problems with the 3-way switches on some early JTV-59 units. Only Line 6 can tell you, based on serial number, whether your guitar qualifies for a replacement switch, and how to go about getting it.