Can a convention be mandatory?

A convention is a common solution to a common problem. The etymology is still a good guide to what it means - con + vent = 'with' + 'come', coming together to solve a problem.

An obvious example is driving on the right versus left sides of the road. Which one gets picked is arbitrary; either will serve. BUT, and it's a big but, everybody has to follow the same convention, or there will be more problems instead of fewer.

When a situation is binary, often one choice has to be made and followed by everyone. The interesting fact is how that choice is made, and how people figure it out. When you get to Australia, you will see signs telling you that people drive on the left side in Australia. When you get to the USA, you won't find signs like that. Customs differ.

An interesting example occurred in Oberlin, Ohio during the 1950s, during the installation of a new dial telephone system. For about a year, some bug in the system caused it to work normally, except that all phone calls ended abruptly after about 5 minutes. People adapted, and since you can get a lot of talking done in 5 minutes, many calls were unaffected.

However, many telephone calls have to last longer than that, so they had to be re-started. The problem was similar to driving on the wrong side of the road. If a conversation between A and B terminated in the middle, both parties would want it to continue; but if they both dialed, the call wouldn't go through. And if both of them assumed the other would do it, the call wouldn't go through either. If A dials and B doesn't, or if B dials and A doesn't, success! But how to proceed?

As it happened, in Oberlin, the telephone subscribers developed their own convention immediately and universally. It was the obvious convention, apparently, because everyone hit on it and understood it immediately; even though it was not logically obvious, it was socially obvious, and that turns out to be the important thing with conventions.

So, what did they decide in Oberlin? If person A called person B in the first place and the conversation was dropped, it was person A who called back. They placed the original call and were assumed to have the motivation. Person B was the passive receiver of the original call, and remained so.

Later on, frequent callers might adopt alternate re-calling for long chats; but the original convention came into place almost instantly, and lasted as long as the telephone bug.

There are two distinct questions here that need to be disentangled:

(1) Does saying 'it is a convention to do X' logically imply 'X is mandatory'?

(2) Is it possible for a convention to be mandatory?

The answer to the first question is no. That it is a convention to do X does not by itself imply that it is mandatory. If some of the OP's colleagues think that it does, they are wrong.

The answer to the second question is, however, yes. A convention can be made mandatory by something outside the convention itself. For example, as has been pointed out in the comments, an employer can make it mandatory that its employees behave in accordance with a certain convention; that is then a mandatory convention within that workplace. The term mandatory convention is thus not self-contradictory.