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Oct 28, 2019 at 17:35 comment added Qayum Khan @MarkGrant : I appreciate the partial answer; however, my specific application (which I didn't specify in the question) fails to satisfy the connectivity hypotheses of your answer here and also the previously linked MathOverflow question. Specifically, all of my finite CW complexes $T, F, E$ are non-simply connected and have homotopy groups in infinitely many dimensions. My opinion is that the results in Ganea's paper are more-or-less a 'thickening' of the preexisting result on Eilenberg--MacLane spaces that he advances.
Oct 28, 2019 at 0:26 comment added skd I would be very interested in knowing the level of generality in which these results hold. Do they work, for instance, in any $\infty$-topos?
Oct 23, 2019 at 18:31 history answered Mark Grant CC BY-SA 4.0