Brake lights don't come on while truck is off

I'm trying to get my vehicle inspection in my state done. The only problem I'm
having is while the truck is off and the brake pedal is pushed down the lights
don't come on. The lights come on while the truck is on. While parked it
comes on, when the pedal is pushed down they come on, and when setting
turn signals they come on. But to be able to pass the brake lights have to
come on while the truck is turned off.

The bulbs have been replaced and the fuse was checked by a mechanic
where I work. He told me the next thing to try would be the switch under the
steering wheel.

I'm just curious whether any one else has experienced this problem and
what the cause is?