Rational-real-part conjecture for cusp shapes of equal-volume fillings

Let $M$ be a $1$-cusped hyperbolic $3$-manifold. A rational-real-part cusp shape means that the cusp shape has rational real part for some choice of basis $\mathbf{m},\mathbf{l}$ of $H_1(T^2)$. Rational-real-part conjecture. If

$$\operatorname{vol} M(p,q)=\operatorname{vol} M(p',q')$$

for distinct Dehn fillings with sufficiently large $|p|+|q|$, then the cusp shape is a rational-real-part cusp shape. The conjecture predicts that sufficiently large distinct Dehn fillings with equal volume can occur only when a suitable change of peripheral basis makes the cusp shape's real part rational. The source derives this conjecture from the preceding cusp-shape norm conjecture and gives experimental examples with rational real parts; no resolution is stated.