I didn't know about CA/Browser forum and the Baseline Requirements. Thanks, will check it out!
// Edit: Relevant section:
The Subscriber Agreement or Terms of Use MUST contain provisions imposing on the Applicant [..] the following obligations and warranties:
[...]
Protection of Private Key: An obligation and warranty by the Applicant to take all reasonable measures to assure control of, keep confidential, and properly protect at all times the Private Key [...]