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Only one line in troubleshooting :
PHP max_input_time: 30 max_input_time: 300 -1 Check your php.ini
but host already answer he can't modify his php.ini (I can't access to this file)
if you dont want keep re-apply that modification, you should asking your hosting company for any out going connection blockers.
maybe IP address or read XML from out site server.
if your host canot modify php.ini for each site, that really bad news. seem you need to changing site hosting. sorry to say that.
only old fashion hosting company cant changing php.ini for each domain.