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The ellipsoidal capacity function of a symplectic four manifoldXmeasures how much the form onXmust be dilated in order for it to admit an embedded ellipsoid of eccentricityz. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions,Xis said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfacesH_{b}formed by blowing up the projective plane with weightb. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parametersbthat are blocked, i.e. have no staircase, and an uncountable number of other values ofbthat do admit staircases. The remainingb-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long asbis not one of these special rational values, any staircase inH_{b}has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations onH_{b}is echoed in the structure of the set of blockedb-intervals.