The correct answer is D, tangible.
The word palpable means able to be touched or felt.
In the sentence, the rival's dislike of one another is so strong it could almost be touched. Of course, their dislike couldn't literally be touched -- however, this is just a way of describing how much they disliked each other.
Of the possible answers, only "tangible" means the same thing as "palpable." The word "tangible" also means able to be touched or felt.
Therefore, the correct answer is D.