Skip down to "Sony ICF-2010 headphone jack repair" at and you see those contacts on the jack. Since this jack is exposed to the world, grim and such can get in there. Do you have a background in electronics repair? If so, you get out the electronics spray contact cleaner and do the work. If that fails you replace the jack.

It's pretty simple but if this is in warranty, you let them fix it.

There is a slim chance that there are PROPERTIES in some control panel to mute those speakers but you would have looked for that. It's not a feature I see as this design is pretty standard. That is, worth looking for but don't count on it.