This is not an answer to the original question, but rather some thoughts about an alternative solution(s) for the overall situation.
I am somewhat bothered by the shared e-mail account approach. E-mail concept, and hence the functionality of the e-mail programs and interfaces are not designed with the multi-user access in mind.
I don't know if you are using Gmail or some other free provider for this address, but those frequently might start complaining seeing concurrent access to the mailbox from different geographic locations.
There are two typical mechanisms for delivering a single message to multiple recipients using a single address: mail aliases on the server (that expand to multiple users) and mailing lists. The former is frequently implemented on the private e-mail servers but not available with Gmail and alike.
The latter can be implemented using Google (or Yahoo) groups.
You can set up and configure a Google group so that only the admin (e.g. you) can subscribe people to that group (i.e. no open subscription), and to keep its archives private or turn off archiving completely. And you can initially subscribe (aka "add members") yourself and the other two people to this group using everybody's individual addresses. In the group settings, you should allow non-members to send messages to the group. Then the group's address (e.g. iceman@googlegroups.com ) can be given to all correspondents.
And then you configure how replies to the messages received by the group members are sent (
https://support.google.com/groups/answer/4600446?hl=en )
This might resolve your original problem.
The problem it might create for you is that responses to the messages will appear from the individual e-mail addresses, not from the group one. But if it is not an issue, - then you don't have a problem.
