DKIM and EXIM: sometimes friends, sometimes not?

Question

Exim

EXIM is configured to use DKIM, the keys are in place, there is a DNS entry. A test email sent from the console to [email protected] returns a positive result (dkim=pass).
An email sent to gmail.com also arrives correctly signed.
However, the email sent by the logwatch process to the same gmail address arrives with the status dkim=neutral, body hash did not verify.
How to fix it?