Which ones do you think are correct?
Girne Castle houses the oldest shipwreck ever ----.
A) having been discovered
B) being discovered
C) to have been discovered
E) been discovered
F) to be discovered
To me, C, D, E and F are OK. Does the presence or absence of "ever" in the sentence play an important role in your choice?
My choice: D); C) and F) are iffy.
C) houses the oldest shipwreck ever to have been discovered.
In affirmative clauses with superlatives [e.g., 'the oldest] ever is used (especially followed by the 'to + infinitive' verb) to emphasize the uniqueness of something or someone:
Neil Armstrong was the first man ever to walk on the moon.
As for choices D) and F), they are variants. 'to be' is optional in F) and recoverable in D).
D) houses the oldest shipwreck ever discovered.
F) houses the oldest shipwreck ever to be discovered.
F) is iffy to me because it can express a future meaning. Which could be the reason 'to be' is omitted in D).
Your choice E), shown here below, is rather awkward. 'to be' can be recoverable in D), but 'to have' cannot be recoverable here,
E) ?houses the oldest shipwreck ever been discovered.
=>houses the oldest shipwreck ever to have been discovered.
E) houses the oldest shipwreck that has ever been discovered.
In short, choices C), D), and F) all house an infinitive. It's recoverable in D), which is why 'ever' works; it's overt in F) but lends itself to a future meaning, and it's overt in C) but why passive voice?
All the best.