tee pipeline killed before the end when signal caught - bash

Question

I have this kind of script program.sh :
#!/bin/bash
catch_sigint()
{
echo "program interrupted"
exit
}
trap catch_sigint INT TERM
catch_exit()
{
echo "program Exiting"
}
trap catch_exit EXIT
#rest of my program...
echo "program Running"
sleep 2
echo "program normal exit"
that is called by another script like that:
#!/bin/bash
catch_sigint()
{
echo "script interrupted"
exit
}
trap catch_sigint INT TERM
catch_exit()
{
echo "script Exiting"
kill -TERM $prog_pid
wait $prog_pid
echo "script END"
}
trap catch_exit EXIT
./program.sh &> >(tee -a log) &
prog_pid=$!
ps $prog_pid
# my code...
sleep 5
echo "script normal exit"
I can see on the terminal and in the log file that my program.sh is "Running". However I can't see that it is exiting. Indeed, when I do not do the redirection to tee, I can well see on the terminal that it is "Exiting", so the kill by pid is working.
So I think the tee redirection is killed at the same time than the program.sh but what I would like is to see the whole logs in the terminal and the log file. How can I do that ?
Update on 13 March 2019
I have updated the code to show the real use case that really show the issue. In fact, each script has its signal trap. So I guess that the tee redirection is interrupted as the rest of the program that's why it shows nothing. My output with interruption:
program Running
PID TTY STAT TIME COMMAND
4922 pts/1 S+ 0:00 /bin/bash ./program.sh
^Cscript interrupted
script Exiting
script END
My output without interruption:
program Running
PID TTY STAT TIME COMMAND
4915 pts/1 S+ 0:00 /bin/bash ./program.sh
program normal exit
program Exiting
script Exiting
./script.sh: ligne 13 : kill: (4915) - Aucun processus de ce type
script normal exit
script END

Answer 1

I modified your 2 scripts mimiking the run time by sleeps and printing the batch process. You can see the "Running" and the "Exiting" from the output. Probably, in your case the calling script kills the program.sh to quickly?
program.sh
#!/bin/bash
catch_sigint()
{
exit
}
trap catch_sigint INT TERM
catch_exit()
{
echo "Exiting"
}
trap catch_exit EXIT
#rest of my program...
echo "Running"
sleep 2
script.sh
#!/bin/bash
./program.sh &> >(tee -a log) &
prog_pid=$!
ps $prog_pid
# my program...
sleep 1
# end of the program ...
kill -TERM $prog_pid
wait $prog_pid
As a result I get:
$ ./script.sh
Running
PID TTY STAT TIME COMMAND
3081 pts/4 S+ 0:00 /bin/bash ./program.sh
Exiting
Hope this helps.