This sentence expresses a conditional - unless x happens, y will happen. The possible future event y depends on whether x happens. Here, x is the conservator finding a way to preserve the photographs, while y is the photographs deteriorating. So the photographs deteriorating is the possible future event. This means we should use the future tense - we can eliminate A and C for using past perfect. (Note: we can also eliminate C for "have been ... deteriorate", which isn't a tense at all - this would only been correct if it were "have been deteriorating and will continue to deteriorate.)
D includes both "continue to deteriorate" and "as they already have". This is redundant. "Continue" tells us that the deterioration has already started to occur, so we don't need to say it again. Eliminate. (Note: we can eliminate A and C for the same reason - there's no need to say that the photographs "have deteriorated" or "have been deteriorating" if we are already saying that they will "continue" to deteriorate.)
E is tricky. It replaces "continue" with "yet more" and changes from simple future ("will continue") to future progressive ("will be deteriorating"). Progressive tenses should only be used if there is a reason - we use them when something else happens while we are doing something. For example, my phone rang while I was running. My dog always barks while I am vacuuming. When the alarm goes off, I will be sleeping. Here, nothing else is going to happen while the photographs "will be deteriorating", so there is no reason to use a progressive tense. We can eliminate E and pick B.
