My personal understanding is this:

"Mr. Li" and "Li Hong" originally operated in cooperation. Later, the two companies parted ways, one called Mr. Li. Both of them should be authentic, but the ownership of trademarks has been controversial for many years. "Li Hong" adopts the way of joining foreign companies, and "Teacher Li" adopts the form of opening a shop directly in order to ensure the quality.

In addition, there are brands such as "Golden Chopsticks" and "Yihetong". Many new stores have been opened recently.

Generally speaking, the operation of "Teacher Li" is more formal and more recognized by consumers. But even for "Mr. Li", the quality of operation between stores is different.

No matter which one you join, whether it is "authentic" or not will ultimately be judged by customers: whoever has delicious beef noodles is authentic.