Using Issue Picker in a public project leads to HTTP error 401

Using Issue Picker in a public project leads to HTTP error 401 when trying to input value to Issue Picker.

• Error message from browser:

   
  Client must be authenticated to access this resource.
  

• No issue suggestion is generated from the dropdown list.
  
• No error in logs as well.

Steps to Reproduce

1. Create a new project named Project A with key: PROJA.
2. Set its Browse Projects permission to a group: Anyone on the web.
3. Remove the existing permission: Application access - Any logged in user.
4. Create an Issue Picker (Single) scripted field and set the JQL to:

    
   project = PROJA
   

5. Open the web browser in incognito mode and try to create an issue in Project A.
6. When selecting the Issue Picker, it hit the HTTP error 401:

   Request URL: http://localhost:8080/rest/scriptrunner-jira/latest/issue/picker?showAvatar=true&showSubTasks=true&query=&currentJql=%20project%20%3D%20PROJA&max=30

   Request Method: GET

   Status Code: 401

Attached the HAR file: error.har

Notes

The error message seems to be coming from Jira and not ScriptRunner which believed to be some Jira security measure placed by Atlassian that wraps around its third-party plugin's REST endpoints.

Not sure if this is something that can be a workaround from the vendor side or has to be done from the Atlassian side.