To put it simply, "no". I can't find a verifiable source from any German that actually states that. Plenty of post-war historians like to tout the claim, and you'll see WW1 veterans on the Allied side saying this in interviews or in their own memoirs.

I'm a big Lee-Enfield fan. Own a bunch of them, shoot them, read about them, and debate on the internet about them. The Mons Machine Gun Myth is unsubstantiated by actual, primary source proof.

Also, just to wrap your head around the fire capabilities of the SMLE (and also a Vickers .303, for fun), here's a link to a thread on /r/guns where some of us sat down and did math on the relative shot times, round counts, etc:

https://www.reddit.com/r/guns/comments/3glvzq/history_and_context_of_heavy_machine_guns_in_the/