Uniqueness conjecture for pitch-one tetragonal and rhombohedral TPMSg3s

Let $\mathcal{T}$ be the family of TPMSg3s with order-4 screw symmetries and let $\mathcal{R}$ be the family of TPMSg3s with order-3 screw symmetries. For a torus parameter $\tau$, write $\operatorname{Re}\tau$ for its real part, and say that $\tau$ solves the period condition for a family with pitch $1$ when the corresponding period equation is satisfied.

Uniqueness conjecture. For every $r\in(-1,1)$, there is a unique $\tau$ with $\operatorname{Re}\tau=r$ that solves the period condition for $\mathcal{T}$ with pitch $1$. For every $r\in(-1,1/2)$, there is a unique $\tau$ with $\operatorname{Re}\tau=r$ that solves the period condition for $\mathcal{R}$ with pitch $1$.

The conjecture would classify the pitch-one members of these symmetry families. Weyhaupt proved the claim for selected parameters corresponding to the gyroid and Lidinoid, but the stated uniqueness for all parameters remains open.