A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_0>H_1>\cdots$ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain $H_{i+1}$ and are contained in $H_i$. This property $𝔓$, say, was investigated in a previous paper by the authors, where soluble groups with $𝔓$ and locally nilpotent groups with $𝔓$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite...