Smooth discrepancy unboundedness under a nonintegral product-of-forms condition

Let $\mathscr L_k$ denote the space of lattices of the form $A\mathbb Z^k$ with $A \in \mathrm{SL}_k(\mathbb R)$, and let $k \geqslant 3$. Let $f$ be the product of the linear forms defined by the rows of $(A^{-1})^T$, and assume that $f$ is not proportional to a polynomial with integer coefficients. Let $\phi:[1,\infty)\to[1,\infty)$ be non-decreasing and satisfy the paper's doubling and lattice Diophantine lower-bound conditions. Let $\boldsymbol{\omega}\in\mathcal G_{k,\ell}$, with $\ell$ satisfying the paper's smoothness condition. Smooth discrepancy unboundedness conjecture.

$$D_{\boldsymbol{\omega}}(\Lambda;N) \text{ is an unbounded function of }N.$$

The paper presents this as a refinement of the two preceding conjectures, using smooth lattice discrepancy to strengthen their expected unboundedness consequence. The supplied excerpt does not establish the refinement.