Concept:The sentence is a third conditional, describing an unreal or hypothetical past situation.
Explanation:The rule for the third conditional is: If + past perfect, then + would have + past participle.
The main clause, "they would have been lynched," already uses "would have been."
So the if-clause must supply the past perfect tense.
Since the word "caught" follows the blank, option D ("had been") completes the past perfect passive phrase "had been caught."
Therefore, the full sentence becomes: "If the armed robbers had been caught, they would have been lynched."
Option A ("would be") is wrong because "would" cannot be placed in the if-clause of a third conditional.
Option B ("have been") is the present perfect tense, which does not fit the past hypothetical meaning.
Option C ("are") is the simple present tense, which belongs to a first conditional structure.
Only option D makes the clause grammatically complete and correct.
Answer:D. had been