I repair the behavior with following lines before the closing exit of the script 'startx':

#---------------------- Script Start ----------------------------------
# get pid of tee and kill it
s=`/bin/ps -C tee -o pid=,args= | /usr/bin/grep -m 1 "xsession-errors"`
err=$?
if [ "$err" -eq 0 ]; then
	s=${s## }
	pid=${s%% *}
	kill -s 9 $pid
fi
#---------------------- Script End ------------------------------------