The conjecture on weak homogeneity and moderate Schwartz pairings

Let $\mathcal G(\mathbb R^d)$ denote the generalized function algebra, and let $\mathcal S(\mathbb R^d)$ denote the Schwartz space. A generalized function $u\in\mathcal G(\mathbb R^d)$ is weakly homogeneous of degree $\alpha$ when it satisfies the weak homogeneity condition of degree $\alpha$. A representative of $u$ is a net $(u_\varepsilon)_\varepsilon$ representing $u$.

The conjecture. If $u\in\mathcal G(\mathbb R^d)$ is weakly homogeneous of degree $\alpha$, then there is a representative $(u_\varepsilon)_\varepsilon$ of $u$ such that, for every $\phi\in\mathcal S(\mathbb R^d)$, the net

$$\left(\int_{\mathbb R^d}u_\varepsilon(x)\phi(x)\,dx\right)_\varepsilon$$

is moderate.

This is proposed as an analogue for generalized functions of the corresponding result for homogeneous distributions. The supplied text does not indicate whether the conjecture has been resolved.