Rolex Trench Watch or Not? ID Help Needed

Hello everyone, I went rummaging through my collection again and was curious to find out more info about this timepiece. Let me just tell you what I already know, and perhaps someone can add some more. It concerns a Rolex with an Aegler Rebberg caliber. The case is known as a trench watch, but the year suggests that this is a timepiece from after WWI. I also think it is a converted pocket watch. The previous owner bought it in Canada but couldn't tell me much more about it himself. It also seems to me that the dial was renewed at some point over the years, as there is no mention of Rolex. I also find it suspicious that the initials of Wilsdorf and Davis cannot be found on the case. If anyone can tell me more, I would love to read it here. Thanks in advance. 911

The engraved date suggests that it missed WWI trench action by almost a decade.

I also had an older timepiece engraved for my son myself, and taking into account the fact that Wilsdorf only used the Rebberg until 1926, I suspect that the production year does not correspond with the engraving. Could the serial number not lead to the year of manufacture?

Your suspicion is reasonable, but there are a couple of important nuances.

Ok. Can you explain that in more detail?