Connection lost with your plugin

Reading through the thread you linked, it looks like the original poster concluded that BLC was not the root cause. They were not 100% certain what caused the problem, but server load and running out of memory were mentioned as possibilities.

Have you tried temporarily deactivating BLC to confirm it was related? Did that fix the problem?

Yes. Deactivating BLC the issue has been fixed and the error message no longer exist (including several issue during the loading of wordpress).

I see. What is the average memory usage like on your site, and how close is it to the limit? I’m wondering if BLC might be pushing it over that limit inadvertently.

You can check memory usage plugins like WP-Memory-Usage or Debug Bar.

Memory limit : 268435456 MB
Memory usage : 41.43 MB

Well, I don’t have any other ideas at the moment.

268435456 MB

You probably meant 268435456 bytes, which is 256 MiB. 268435456 MB would be 256 TB.

Memory overview:

PHP Version : 5.4.23 / 64Bit OS
Memory limit : 268435456 MB
Memory usage : 41.54 MB

As the author of the post linked to above I should point out I have just updated it, and, in short, I have had and am having no problems with BLC.

However, BLC is more resource-intensive than other plugins I am running and it may be, if it is the case that the newer versions of WordPress themselves are also more resource-intensive than previous versions, that what is occurring is excessive competition among the parties involved for limited memory resources.