Mathematica does not distribute conjugation when used with Assuming sometimes?

Question

If I input:

Assuming[$x_{-} \in$ Reals, FullSimplify[Conjugate[$e^{i x_1}x_2$]]]

Then the output is:

$e^{-ix_1}x_2$

As it should be. And if instead I do:

Assuming[$x_{-} \in$ Reals, FullSimplify[Conjugate[$e^{ix_1}x_2+x_3$]]]

The output is:

$e^{-ix_1}x_2+x_3$

Again as it should be. Now for some reason if I do:

Assuming[$x_{-} \in$ Reals, FullSimplify[Conjugate[$e^{ix_1}x_2+e^{ix_3}x_4$]]]

The output is:

Conjugate[$e^{ix_1}x_2+e^{ix_3}x_4$]

So for some reason mathematica does not distribute conjugation over addition in this last case. Mysterious and annoying, as I really need it to in order to progress with writing my code.

Answer 1

Like the OP, I find that

Assuming[x_ ∈ Reals, FullSimplify[Conjugate[Exp[I x1] x2]]]
(* E^(-I x1) x2 *)

works, but

Assuming[x_ ∈ Reals, FullSimplify[Conjugate[Exp[I x1] x2 + x3]]]
(* E^(I x1) x2 + x3 *)
Assuming[x_ ∈ Reals, FullSimplify[Conjugate[Exp[I x1] x2 + Exp[I x3] x4]]]
(* E^(I x1) x2 + E^(I x3) x4 *)

return the wrong answer without explanation. (Note that the Question shows only the last of these giving the wrong answer. It may be that, as suggested by Sjoerd C. de Vries, patterns do not work with Assuming. So, we try instead

Assuming[(x1 | x2 | x3) ∈ Reals, FullSimplify[Conjugate[Exp[I x1] x2 + x3]]]
(* E^(-I x1) x2 + x3 *)

which gives the correct answer. Finally,

Assuming[(x1 | x2 | x3 | x4) ∈ Reals, FullSimplify[Conjugate[Exp[I x1] x2 + Exp[I x3] x4]]]
(* Conjugate[E^(I x1) x2 + E^(I x3) x4] *)

also gives a correct, although not useful, result. Of course,

ComplexExpand[Conjugate[Exp[I x1] x2 + Exp[I x3] x4]] // TrigToExp
(* E^(-I x1) x2 + E^(-I x3) x4 *)

works fine.