Question: 
When I have Duplicate Check and Plauti Cloud open on two tabs in the same browser, and I switch between the two, I get errors stating ‘error parsing json response: "unexpected token ‘<’’. logged in?" and then I am logged out of Salesforce.

Answer:
Refreshing the Duplicate Check tab should solve this. If not: either open Duplicate Check and Plauti Cloud in two different browsers, or close the Duplicate Check tab before opening Plauti Cloud.