Cyclotomic Eisenstein conjecture for S-arithmetic homology

Let $T$ be a finite set of primes with $|T|=d$, let $K[1/T]$ denote the corresponding $S$-arithmetic level, and let $\mathcal{F}$ be the coefficient system used in the paper. Cyclotomic Eisenstein conjecture. Then $H_i(K[1/T],\mathcal{F})$ is cyclotomic Eisenstein for every $i\leq d$. The conjecture is a refined form of the low-degree Eisenstein-homology principle and would force the localized spectral sequence at a non-Eisenstein ideal to converge to zero in the range $p+q\leq d$.