So here is the solution I found thanks to our brainstorming:
<?php
$file_data = file_get_contents('/home/MYFILE.csv');
//$utf8_file_data = utf8_encode($file_data);
$utf8_file_data = mb_convert_encoding($file_data, "UTF-8", "UTF-16LE");
//$utf8_file_data = iconv("UTF-16LE","UTF-8",$file_data);
$new_file_name = '/home/MYFILE_NEW.csv';
file_put_contents($new_file_name , $utf8_file_data );
?>
The only pb is that the output size is twice as big as the input. If I use ICONV on my PC it is HALF the size...
If anyone knows I'd like to hear why.