Following are the steps I did on your demo frontend (Demo 2 with geo user’s IP address):

1. Enter the front page, the currency showing JP dollar for example. (correct)
2. I choose British pound and click "change currency".
3. The demo store price currency showing £. (correct)
4. I choose a product and add it to the cart.
5. I click "show cart" and link to the cart page.
6. The cart page show JP dollar. <-- (Why?)

My question is, can the currency stay the same after I click "change currency" button?
In this case, it should be still showing "Britsh pound" on the cart page.
(Of course, only the "Total in Payment Currency" can remain the same, in your case, is Eurodollar € .)

Thank you.