Linux: process of a bash shell launched by crontab still running after the shell is terminated - linux

Question

There is an issue I would like to solve: I'm going to deploy on a Linux Red Hat 5 production environment (called PENV) a php web application under apache server; I am developing such this application on a development environment (called DENV) with Linux Mint 20.3.
On DENV, I created a crontab for the user www-data containing the following scheduled command:
0 4,12 * * * sh /bdir/s_etlShell.sh >/dev/null 2>&1;
the shell /bdir/s_etlShell.sh starts everyday at 4.00 AM and at noon, and its execution lasts between 2 and 10 minutes. It also writes to a logfile /bdir/logshell.txt.
the last two instructions of the shell are
echo "SHELL TERMINATED" >> /bdir/logshell.txt
exit
Past the 4.00 AM and noon, I found SHELL TERMINATED as a final statement inside /bdir/logshell.txt, but when I give the following command by terminal
ps fax | grep "s_etlShell.sh" | grep -v grep
I get the following output (the PID's are varying obviously):
ps fax | grep "s_etlShell.sh" | grep -v grep
1596 ? Ss 0:00 \_ /bin/sh -c sh /bdir/s_etlShell.sh >/dev/null 2>&1
1605 ? S 0:00 \_ sh /bdir/s_etlShell.sh
the processes of the shell look as if they were still active despite the shell terminated. I would expect no output instead.
I need to check the status of the shell execution in the web application via php script (check_etl_shell_status.php) launched every 2 seconds by the following javascript funcion
function loadCall() {
setInterval(function () {$("#id_content").load("check_etl_shell_status.php",'q='); }, 2000);
}
the function loadCall() is being called on load the home page.
The content of check_etl_shell_status.php is the following
<?php
$output = shell_exec('ps fax | grep "s_etlShell.sh" | grep -v grep');
if ($output) {
echo "shell is still running...";
} else {
echo "shell terminated";
}
?>
and the output message is displayed inside a div of the home page
...
<div id="id_content"></div>
...
is there a way to make sure that, when a shell has terminated, whether is launched by crontab or on demand by web application, I have the right information on its status?
Thanks to whoever can help me