When a customer adds some products to his/her cart, it generates automatically a quotation in the backend that will remain open. That quotation will be used again to add future products inside when the customer adds more products to his/her cart.

When the product is removed from the cart, the quotation still exists but is empty. Note that the pricelist is still the one you set previously on the customer.

Let's imagine that for whatever the reason, you decide to set another specific pricelist to the same customer 

On the shop, the customer will still see the price related to the first pricelist set.

All that is due to the quotation still present in the backend with a pricelist still set for 10$

It will still be 10$ if I go through the entire purchase process on the e-shop as well because the product will be added to that same quotation.

At the moment, the only way to make the new pricelist work as expected on the website is to make sure you don't have any quotations open with no products inside and still related to the previous pricelist. If you delete that one, you gonna see that the new pricelist will be applied correctly.