Weak Herstein–Scott conjecture for subnormal subgroups

Let $D$ be a division ring and let $D^*$ be its multiplicative group. A subgroup is non-central if it is not contained in the center of $D$. Weak Herstein–Scott conjecture. Every non-central subnormal subgroup of $D^*$ contains a non-central normal subgroup of $D^*$. This is presented as a weak version of the Herstein–Scott conjecture and is attributed to Parker and G. in the source; no resolution is given here.