Conjecture on modular identities relating two parameter values

Let $q=e^{-\pi t}$ with $t>0$, and let $z(t)$, $\tau(t)$, and $k(t)$ be the functions used in the source. Write $\tau_j=\tau(t_j)$ and $k_j=k(t_j)$ for $j=1,2$.

Modular-identity conjecture. If one of the relations

$$z(t_2)=z(t_1),$$ $$\tau_2=\tau_c^2 \cdot \frac{4\tau_1}{4\tau_1^2-k_1^2},\qquad k_2=-\tau_c^2 \cdot \frac{4k_1}{4\tau_1^2-k_1^2},\qquad t_2=t_1\cdot\frac{2\tau_1-k_1}{2\tau_1+k_1}$$

holds, then all the other relations hold as well.

The text supplies no proof or resolution status for these identities; the surrounding material presents them as a new conjecture and reports numerical support for the related calculations.