Mine does turn off and it drives me bonkers
I sometimes leave the stylus on the table for a long time, while reading a text. And when i grab it to jot something down, i often find the digitizer went to sleep and i have to either put the stylus in the silo then pull it out again or click the buttons around for a while, until iLiad finally grasps what it should do (i.e. switch the bloody digitizer on).