Elixir sent me a replacement set some two weeks ago. This is what Elixir answered me by e-mail (I am not sure whether I am reassured...):
Dear Karel,
Thanks for your interest and support in Elixir® Strings, along with taking the time to write us. I apologize for the experience you are having with our strings.
While it is not out of the ordinary for our coatings to fray, normally it happens over an extended period of time.
The frayed coating will not affect tone and the strings can remain bright long after the strings begin to fray.
Acoustic and electric guitar players typically see little particles or flakes in the areas that get the most abrasion.
Also comparatively heavy picks, specially out of metal, may prematurely cause damages on the coating.
While not pretty, the worn coating does not impact the string tone as long as the fingerboard is in a good condition.
Because there the strings are more often in contact with sweat and dirt and were well protected as long as the coating is intact.
Thanks and kind regards,
Elixir® Strings