How to control the screenshots of my website in the recently used websites list shown on the new tab in my browser

Question

Browser like Firefox and Chrome take screen shot of the visited websites and can show them on a new tab as "recent used website".

Since my website is showing confidential information, how could I avoid the browsers to take screen shots for the "recent website" list, or at least limit it to the login page (like Facebook or banking website are doing)?

I found a possible answer in Is there a W3C standard meta tag to determine the cover image used to represent a website? but it seems a still unstable method, and banking sites I investigated are not using it apparently so I suspect there is another (better?) solution.

Any idea? I need it working at least for Firefox.

Note: sites are using HTTPS. Under Chrome, use HTTPS seems to solve it since blank screen is shown as preview. Firefox does show it even when using HTTPS.

UPDATE: On Mozzila's support page https://support.mozilla.org/en-US/kb/thumbnails-new-tab-page-missing-how-get-them-back they state the following:

Note: Some websites don't allow images (--> thumbnail) to be generated and saved

Unfortunately they don't tell how to do it... But yes it seems there is a way to avoid it...

UPDATE 2: What I actually am looking for is that the site uses cache for Javascript and images, but not for thumbnails. Use of HTTPS would to solve it, except because of the Firefox bug.