Hi there,
I did try many tests, by duplicating the post and adding the template for the homepage and using the WPML and changing the settings of the WPML.
Unfortunately, I could not find why this is happening on your page. There should be a problem of some kind in the content of the homepage itself and not the fact that it is the homepage, cause I changed the front page for a moment to another page and tested the whole process and it again fails.
Maybe it is related to the fact that the homepage is pretty large one and it fails to allocate enough memory, but then you said you even deleted all the sections and it does not save again.
I did another test and had an interesting result. I added a template to your homepage and downloaded it to my local machine. I added it to my local installation and it worked ok and I could translate it to Armenian with no issue in the Cornerstone.
So this is indeed related to your installation and server environment but unfortunately, I can not find the cause. The last thing that I can do is to give you my local installation properties and hope you can make your environment like mine and see if it works for you:
My WPML version: 3.8.3
My server settings:

Thank you.