The dream is a test applied to both brothers. Not a test relating to the Ring Quest but a test relating to the King Quest.
The first words of the dream are:
The words are a command to the sons of the Steward to seek their king. It comes "first" and then "oft" to Faramir because he is the most receptive; does he not place the re-establishment of the kingship first when he speaks what I call his "affirmation" to Frodo? He does.Seek for the Sword that was broken
In Imladris it dwells
Boromir fails the King test then and there in Minas Tirith, even though he doesn't understand it, because - subconsciously - he does not want the king to return. And yet, because of his pride he insists on taking on the mission to Imladris."I would see the White Tree in flower again in the courts of the kings, and the Silver Crown return, and Minas Tirith in peace."
This is why so much stress is laid on the length and difficulty of Boromir's journey: "a hundred and ten days I have journeyed all alone", and "long have I wandered by roads forgotten." When he arrives his cloak is "stained with long travel".
Faramir confirms:
And Gandalf:"I should have been chosen by my father and the elders, but he put himself forward, as being the older and the hardier (both true), and he would not be stayed."
I note that Boromir's first words to Aragorn are questions: like Thomas he does not recognise his King:"Boromir claimed the errand and would not suffer any other to have it. He was a masterful man, and one to take what he desired."
his contrasts with Faramir's immediate, Biblical recognition/affirmation of Aragorn's kingship:"And who are you, and what have you to do with Minas Tirith?"
Just my interpretation."My lord, you called me. I come. What does the king command?"
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