@MandeepRawat please carry out a PSC key word search first to check on existing posts about this. I found the following:

 Not back to home when close case in new tab 

Developing applications that open in multiple browser tabs or browser windows

This issue has been noted in similar contexts. For example, in Cosmos, when a case is finished or closed in a new tab, the UserPortal may become blank instead of returning to the Home page. This behavior is expected in some versions. A feedback request (FDBK-76396) was created for review by the Product team. As of version 8.6, multiple tab/window functionality was improved to support opening cases in new browser tabs without refreshing one another. You might want to check if you are using the latest version and if the issue persists, consider updating or reviewing the configurations

---> Since cosmos is Single Page application, this is expected behaviour from Pega where the current thread is being retained and previous thread is not retained in clipboard.

A feedback request (FDBK-76396) was created for the same for review of this requirement by Product team. However, note that as of 8.6 the multiple tab/window functionality was improved. In 8.6 onwards we support opening cases in new browser tabs. 

Enhance the case processing experience in Theme Cosmos applications (8.6)

" Whether you start at the home page, follow a link, or use bookmarks, new tabs or windows no longer refresh one another. As a result, you can keep referencing or cross-checking several work items at a time without sacrificing one context for another and losing current work. Additionally, table settings are maintained for the duration of the session. For example, if you apply filters and sorting to a list, and then open a new document, the application retains the table settings when you navigate back to the page by clicking the browser back button."

Suggestion:

 https://community.pega.com/support/support-articles/case-displays-new-window

https://collaborate.pega.com/question/open-case-new-browser-tab

Also see the following which explains Cosmos-recommended navigation of cases in tabs: https://community.pega.com/video-library/pega-84-update-whats-new-cosmos-design-system

I found a possible workaround:

Remove the Home action added on the button and configure the Launch Harness and add the pages and Harness according to the requirement.

Below is the screenshot of the same implementation for your reference.

Note: The Harness that is added(Home) is the desired Harness where the page should be navigated to. 

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Other posts:

How to get "Open link in new tab" option on right click

How to open an assignment in a new browser tab using Theme-Cosmos

Can you include screenshots of your issue?  What is the control where you are clicking to open the case? 

You may need to run a browser/ network trace and log this issue as a support incident via the MSP. Please provide the INC id here.