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What is special about the Pouch of Douglas is that it is the lowest point of the peritoneal cavity, so even if blood gets in between the layers of the broad ligament (which I agree with you, it looks like it should), I bet it somehow makes its way down to the Pouch of Douglas due to gravity.
Bless you for including the spoiler warning
Actually I think @hello is right.
First, because that's the only way this shitty wording question makes sense, second cuz it says "that lies on the EXTERNAL SURFACE" that means outside of the spermatic cord, and the genital branch of GF nerve actually does:
The genital branch of the genitofemoral nerve enters the inguinal canal via the deep inguinal ring. (https://www.ncbi.nlm.nih.gov/books/NBK430733/), and then exits to innervate the anterior and superior scrotal skin (also the cremaster muscle).
CORRECTION: Nevermind, everything I said is wrong. I hate this question