Correct, but using persent perfect with already is more natural in formal English, since simple past is always tied in the speaker's mind at least to a specific time in the past. Here it would most likely be some unspecified or uncertain time in a period of time before now.
A bit nasty to ask suddenly for a past time, especially when the pat and the present have the same form .. in a lesson about past. In many other sentences in which present was un-natural the present was to be used.
Since this is supposed to be present tense exercises, I put "I think we are already crossing the border." Would that be possible? I agree that the verb is more likely to be understood as past, but does it have to be?
That's the key, as in Spanish. When the verb in present and preterit is identical, the subject matter and the já are cues that the meaning is past. As in English we can tell whether "read" is present or past by some time indicator in the sentence.