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do you have any core customization? if not can I download your site from Akeeba Bakcup to make local copy? I need your permission first. Deep debugging at locally more safe, because can damaging the site.
No, no one core customization.
And yes, you can downloads it. Just remember the problem is with my development site and not with the live site.
Please, see the development information on my first post.
at local copy, there is no issue with cron job. I think this issue probably server site issue. please execute the cron job and have a look at server error logs. it must some error message at there.