Excellent episode – I was certain it would be as this is a favorite poem of mine… and I loved the whole discussion of the Hamlet resonances in Prufrock.  
The question of what the unspecified question is is an interesting one with which I’ve wrestled since coming across Prufrock yeargs ago.  I was in the “it’s a sexual question, certainly” camp for a while, but I’m not entirely sure that works.  At some point I settled on another seemingly likely possibility that I’m curious about how you’d respond – that, is, I wonder if the question which is referred to in the poem is in fact the famous question of Hamlet’s soliloquy (in all its fullness, not merely the opening “To be or not to be?” but its fleshing out in the full soliloquy).  The denial of the protagonist that he is Hamlet is an interesting denial, given that there are so many parallels between the narrator and the Prince of Denmark.  Prufrock isn’t comfortable in his own skin – he’s indecisive and lacks confidence in who he is.  I am reminded by Prufrock’s waffling and painful reflections on himself to me seem strongly to resemble Hamlet’s internal questioning in the play, and particularly in the soliloquy.   Death’s mocking of Prufrock, too, reminds me of Hamlet’s musing on death and what lies beyond.  I don’t want to dismiss entirely the notion that the question is connected to the woman who settles the pillow by her heard – perhaps a question like “do you love me?”  that Prufrock doesn’t really want answered, as he’s not sure he’d like more to hear and deal with a positive or negative answer – but this too brings forth something akin to what I think is part of the emotion of Hamlet in the soliloquy, namely, the eternal balancing between two opinions or answers, never fully being comfortable with the unknown possibilities.  Anyway – just my own probably miscalculated musings.  
Fun stuff, gents.  Thanks again as always.