The exact Hausdorff dimension formula for trajectory sets of equivalent $g$-templates

Let $g$ be a general $1$-parameter flow, let $X_n$ be the associated space of lattices, and let $Y_{\Lambda,\mathcal{F}}$ be the trajectory set determined by a lattice $\Lambda\in X_n$ and a set $\mathcal{F}$ of $g$-templates. Write $d_H$ for the metric on the relevant horospherical subgroup $H$, let $\Delta_0(f)$ denote the initial quantity associated with a template $f$, and let $\overline F$ be the dimension function appearing in the Hausdorff-dimension bounds. Assume that $\mathcal{F}$ is closed under equivalence.

Exact dimension conjecture. For every $\Lambda\in X_n$,

\dim_{\text{\it\em\fontfamily{qcr}\selectfont\mathbf{f} H}}(Y_{\Lambda, \mathcal{F}};d_H)=\overline F\left(\sup_{f\in\mathcal{F}}\Delta_0(f)\right).

This conjecture would give an exact, rather than merely bounding, formula for the standard Hausdorff dimension of trajectory sets for general $1$-parameter flows. It is motivated by the known sharp calculation in the three-dimensional example discussed immediately before the conjecture; the supplied text does not state a resolution in general.