No, it cannot.  Evercontact requires access to All Mail to be able to analyze email signatures and then compare your contacts' most recent info with what was there before.

How to make sure your IMAP AllMail is active

1) Click here to go to your gmail label settings  and make sure "Show in IMAP" is checked for both

• All Mail
• Sent Mail 
• NOTE: If you have more than one gmail account, make sure you are in settings for the email address configuration you need to change.