Do you mean that the presence of FVA etc should have an impact on the optimal exercise? Well Andrew Green has a paper on that on Risk, but I tend to believe that typically the optimal exercise is treated separately w.r.t. all xVAs like in the old single curve world...
At least, this seems to me a picture of what is happening in order to keep things short and simple.
Exactly, the thing is that without funding, it is never optimal to exercise whether it is a put or a call, hence it can be priced as a european. With funding it is a totally different story.... Even if we forget about FVA, and assume that we can lend/borrow at a theoretical risk free rate, the future option still cannot be priced as a european. In that situation , it is hard to capture the pure dynamics of the future rate from eurodollar option . I am going to find the paper you are talking about.