I'll be glad to take a look. I logged in and checked the Listings Layout and found this code I assume you are referring to:
[wpv-conditional if="( '[wpv-post-author]' eq '[wpv-current-user]' )"]
[toolset-edit-post-link layout_slug="edit-listing" class="edit-post-link"](edit)[/toolset-edit-post-link]
[/wpv-conditional]
[wpv-conditional if="( '[wpv-user]' eq 'jobsnvf' )"]
[toolset-edit-post-link layout_slug="edit-listing" class="edit-post-link"](edit)[/toolset-edit-post-link]
[/wpv-conditional]
[wpv-conditional if="( '[wpv-user]' eq 'nomad411' )"]
[toolset-edit-post-link layout_slug="edit-listing" class="edit-post-link"](edit)[/toolset-edit-post-link]
[/wpv-conditional]
Anytime you want to troubleshoot conditional HTML, the first step is to add the debug="true" parameter like this:
[wpv-conditional if="( '[wpv-post-author]' eq '[wpv-current-user]' )" debug="true"]
Please add this parameter to all 3 conditional shortcodes.
Next, I can see the only content inside these conditionals is the toolset-edit-post-link shortcode. Please add some basic text content inside each conditional to help during testing, like this:
[wpv-conditional if="( '[wpv-post-author]' eq '[wpv-current-user]' )" debug="true"]
Author edit post conditional: [toolset-edit-post-link layout_slug="edit-listing" class="edit-post-link"](edit)[/toolset-edit-post-link]
[/wpv-conditional]
Now even if there is a problem with the edit post link shortcode, some content will be written to the page if the conditional is true. This will be helpful to determine the issue(s). Add text information like this to each conditional, then test again. You will see some information printed to the screen. Copy the results and debug information written to the screen for me to review.
PS: Do you still offer those shared desktop sessions?
Yes, you can request a Hangouts-based video call here: https://toolset.com/toolset-support-policy/ask-support-video-call/
OR I could record a screencast if needed.. It probably wouldn't hurt
A screencast is always appreciated, and may be enough to solve the problem without needing to schedule a video call. It takes some time to get those approved and scheduled, and that can cause delays in providing solutions.
PPS: I've tried doing a Duplicator backup but I think WPEngine must block it, it fails
WPEngine has some strict policies regarding duplication software. It's usually more effective to create a database dump file and backup your theme and plugins directories. Then provide a download link where I can find a zip archive of all those materials, and I can work from those instead. I don't think it's necessary just yet, let's go through the main troubleshooting steps first and I'll request that info if it's necessary.