Lower bound conjecture for generalised divisor sums

Let $K$ be a number field, fix an ideal class representative $u\in\mathcal{I}_K$, let $f\in\mathcal{Z}_K$, and let $\mathfrak{F}$ be a system of binary forms as in the paper. For an $\mathfrak{F}$-admissible triplet $\mathcal{P}=(\mathcal{D},(\sigma,\tau),\mathfrak{W})$, let $D(\mathfrak{F},f,\mathcal{P};X)$ denote the associated generalised divisor \sum, and let $\rho(\mathfrak{F})$ be the exponent defined for the system of forms. Lower bound conjecture for generalised divisor sums. There \exists a finite set $S_{\mathrm{bad}}=S_{\mathrm{bad}}(\mathfrak{F},f,\nu)$ of \prime ideals of $\mathcal{O}_K$ such that, whenever $\mathcal{P}$ is $\mathfrak{F}$-admissible and $\mathfrak{W}$ is divisible by every $\mathfrak{p}\in S_{\mathrm{bad}}$, one has

$$D(\mathfrak{F},f,\mathcal{P};X)\ggg X^2(\log X)^{\rho(\mathfrak{F})}\quad\text{as }X\to\infty.$$

The implicit constant may depend on every parameter except $X$. The claim is presented as a lower-bound prediction for these generalised divisor sums; the surrounding discussion notes that its special case for constant $G_i$ over $\mathbb{Q}$ is familiar, but says that the result had not appeared in print. Its resolution is not specified in the supplied text.