Coarse total vertical perimeter conjecture on the Heisenberg group

Let $\mathbb H(\mathbb R)$ be the real Heisenberg group, let $\mu$ be its Haar measure, and let $\mathrm{PER}(A)$ denote the horizontal perimeter of a measurable set $A\subseteq\mathbb H(\mathbb R)$. For $t>0$, write $v_t(A)$ for its vertical perimeter, and for $\varepsilon\in(0,1)$ define the coarse total vertical perimeter by

$$V^{(\varepsilon)}(A)=\int_\varepsilon^1\frac{v_t(A)}{t^{3/2}}\,dt.$$

Coarse total vertical perimeter conjecture. For every measurable $A\subseteq\mathbb H(\mathbb R)$ and every $\varepsilon\in(0,1/2)$,

$$V^{(\varepsilon)}(A)\lesssim\sqrt{\log(1/\varepsilon)}\cdot\mathrm{PER}(A).$$

This is a weaker coarse variant of the preceding isoperimetric question. The source reports significant partial positive evidence, including numerical experiments and proofs of nontrivial special cases, but does not establish the conjecture.