The phrase "By the fifth sequel, the director had become $\qquad$ in his delivery; even the most die-hard fans of the series lost interest" implies that the director's work became so uninteresting or uninspired that even dedicated fans gave up on the series.

We are looking for words that describe something uninspired, routine, or lacking in effort, which would cause fans to lose interest.

Let's evaluate the options:
- A. convoluted: Intricate and complicated. While this might make something hard to follow, it doesn't necessarily imply a lack of interest in the sense of being boring or routine.
- B. cautious: Careful and hesitant. This doesn't directly imply a loss of interest due to being uninspired.
- C. remedial: Intended to improve or correct. This doesn't fit the context of a director's delivery in a sequel.
- D. perfunctory: (of an action or gesture) carried out with a minimum of effort or reflection. This strongly suggests a lack of enthusiasm or care, which would lead to a loss of interest from the audience.
- E. unpleasant: Causing discomfort, unhappiness, or dislike. While the movies might become unpleasant, "perfunctory" or "mundane" describe why they became unpleasant for fans.
- F. mundane: Lacking interest or excitement; dull. This perfectly describes something that would cause even die-hard fans to lose interest due to being uninspired or boring.

Both D. perfunctory and F. mundane convey the idea that the director's delivery became uninspired, routine, or dull, leading to a loss of interest from the audience.

The final answer is $\(\mathrm{D}, \mathrm{F}\)$