Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

For an odd prime $p$, let $A_j$ be the $ω^j$-eigenspace of the $p$-primary class group of $\mathbb Q(ζ_p)$. Fix an even integer $d\ge4$, put $N=(p-1)/d$, and let $U_d=(\mathbb Z/d\mathbb Z)^\times$. For a relative density-one set of primes $p\equiv d+1\pmod{2d}$, we prove that the odd block $\bigopl