The root conjecture for frame lattices corresponding to lattices $T_{[abc]}$

Let $T_{[abc]}$ be a lattice in the notation of the paper, and let $W$ be a frame lattice corresponding to $T_{[abc]}$. A root is a lattice vector of minimal norm as defined in the paper. Root conjecture. Every frame lattice $W$ corresponding to a lattice $T_{[abc]}$ contains a root.

This conjecture would answer the dissertation's question affirmatively by ruling out rootless frame lattices of this form. The surrounding discussion uses discriminant-group computations and embeddings of scaled Niemeier lattices into the Leech lattice to make the claim plausible; no resolution is given here.