Local cache leakage between different orgs in Web Console

**Describe the bug**
After using Web Console in one browser in one Salesforce org, when opening Web Console in a different Salesforce org with the same browser, there are some instances of cached data from the first org showing up in the second's Web Console.

**To Reproduce**
Steps to reproduce the behavior:
1. In org A, open Web Console, go to Org Browser, download all Apex Classes, then open a few of those Apex Classes, close them and close Web Console.
2. Using the same browser, in org B, open Web Console, and press Ctrl-P.
3. Notice that some of the Apex Class names you opened in org A are showing in Web Console in org B. The files do not exist in this editor and cannot be opened, but nonetheless they appear unexpectedly.
4. Another example: if the last thing you did in org A was attempt a global search for some string, when you open Web Console in org B, that same string may appear in global search.

**Expected behavior**
Web Console should behave as completely isolated between different orgs. This is particularly important for partners who need to work with multiple clients and avoid interferences between them.

**Screenshots**
In the screenshot below you can clearly see that a class named `AC_InquiryService` is showing up in the Ctrl-P menu, however no such class name exists in this editor running in the context of a dev org. It is in fact a class name belonging to a customer org in which I'd opened Web Console earlier today in the same browser.

**Desktop (please complete the following information):**
- OS: Windows 11
- Browser Firefox, Chrome
- Version Latest versions as of May 1 2026

**Additional context**
This is an issue in both live Prod orgs and pre-release dev orgs.