The distribution of ℓ^{∞}-Selmer groups in degree ℓ twist families II

We continue the investigation of the distribution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Selmer groups in degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> twist families of Galois modules over number fields begun by the author in [J. Amer. Math. Soc. 39-1 (2026), pp. 1–72]. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Selmer groups for a given degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.