The Merel-unit and cubic-field unit conjecture

Let $K$ be a cubic extension with negative discriminant $-D$, let $L$ be its sextic Galois closure, and let $g$ be the associated weight one form of level $D$. Let $u\in\mathcal{O}_K^*$ be a unit, let $q\equiv1\pmod p$ satisfy $\left(\frac{-D}{q}\right)=-1$, and assume $p\geq5$. Let $\varpi_{\mathrm{Merel}}\in(\mathbb{Z}/q)^*$ be nonzero modulo $p$ after projection to $\mathbb{F}_p\langle1\rangle$, and let $\eta$ be the first Fourier coefficient of the Eisenstein component of the projection of $g(z)g(qz)$ to level $q$. Let $\bar u\in(\mathbb{Z}/q)^*$ be the reduction of $u$ modulo the unique degree-one prime of $K$ above $q$. The Merel-unit conjecture. There exist $A,B\in\mathbb{Z}$ such that, for all such $q$,

$$\varpi_{\mathrm{Merel}}^{A\cdot\eta}=\bar u^B\quad\text{in }\mathbb{F}_p\langle1\rangle.$$

The coefficient $\eta$ is well defined modulo the numerator of $(q-1)/12$, which is sufficient for this relation. The supplied text presents this as a rephrasing of the paper's conjecture and gives no resolution.