OPTION 1


If you're using a cocamail address then there is no inbox so there's nothing to read.
If you use an external email address that's forwarded to your character then yes, Conducttr will react and respond as normal and you'll be able to see incoming emails in the external inbox. Be a little careful with this approach as emails that go into spam of course won't be forwarded to Conducttr

OPTION 2


You can parse all the content from the incoming email and re-post it to yourself. The formatting won't look nice but you will at least be able to read what people are sending the character.

Recipe
  1. Give your character a BLOG account and use your own email address as the blog destination. It's not really a blog of course, it's your email
  2. Create an email trigger with this matchphrase
    |parse('all_chars') into this_email|
  3. In the same beat, create a blog post that goes to your email address and in the body write this
    |this_email|
 
What will happen is everything in the email is stored in the dynamic variable "this_email" which is then printed in the blog post when it's sent to you.
In the blog's subject you could write 
Email for My Character from |audience.sender.name| |audience.sender.lname|

You'll then get emails with a subject including the audience' name and last name.