Discreteness of the minimal-displacement spectrum

Determine whether, for every automorphism φ of the relevant free product or free group, the spectrum of possible minimal displacements spec(φ) is a discrete subset of the real numbers, rather than merely a well-ordered subset.

Background

Theorem 7.2 proves that the global simplex-displacement spectrum, and consequently the spectrum spec(φ) of minimal displacements associated with a fixed automorphism φ, is well ordered. Thus every decreasing sequence in the spectrum is eventually constant, but well ordering alone does not establish that the spectrum has no accumulation points from other directions.

Immediately after this result, the paper raises the stronger possibility that spec(φ) is actually discrete. Establishing discreteness would sharpen the spectral information available for displacement values and could yield stronger finiteness or algorithmic consequences.