Is there a chance that using the Facebook like button will open the slide image instead of a random image of the webpage?
When sharing a link on Facebook, there is not a way to dynamically specify an image for the Facebook post. Facebook has its own way to define the post image.
Facebook allows you to use "Open Graph Markup" to define the post image. You can view the document: https://developers.facebook.com/docs/sharing/webmasters#markup. If you don't define the markup, Facebook will find an image from your webpage and use it.
The result is also cached by Facebook, you can use the tool provided by Facebook to force update it: https://developers.facebook.com/tools/debug/.