Possibly, but I doubt it.   Wire under the dash is usually well
protected and doesn't fail.  Wire usually doesn;t fail unless it;s
exposed to a harsh environment like heat, UV rays, etc. that cause the
insulation to fail and a short.    Then it should blow a fuse.

I'd suspect a loose connection, bad switch, or that whatever generates
the current to commmand the change over isn;t generating it.   In
general, with an air circulation flap, that command can depend on one
or more conditions from another switch, sensor for vacuum, temp, etc
that is totally unrelated to the wire connected to the valve.    You
really need to trace a diagram that shows the circuit and what drives
it.