= Solution
Let $H_0^1$ be the <Cameron-Martin space of Wiener measure> on $[0,T]$: its elements start at zero, are absolutely continuous, and have square-integrable derivatives. For $h\in H_0^1$, let $y^h$ solve the controlled <ordinary differential equation>
$$
\dot y_t^h=\sum_{i=1}^d V_i(y_t^h)\dot h_t^i,\qquad y_0^h=y_0.
$$
The <Stroock-Varadhan support theorem> in the uniform topology is
$$
\boxed{\operatorname{supp}\mathcal L(Y)=\overline{\{y^h:h\in H_0^1\}}^{\|\cdot\|_\infty}\subset C_{y_0}([0,T],\mathbb R^e).}
$$
One may equivalently take piecewise linear or smooth controls starting at zero. The skeleton uses the displayed vector fields, as appropriate for the Stratonovich equation.
Choose $2<p<3$ and work in the complete separable space of anchored <geometric p-rough paths>, with its <rough path metric> $d_p$. The <enhanced Brownian motion> is a random element of this space. Its <support of a measure> is
$$
K=\overline{\{S_2(h):h\in H_0^1\}}^{d_p}.
$$
We explain this Brownian support step using the permitted results about the enhancement. The almost-sure polygonal convergence proved in Question 3 gives the inclusion of its support in $K$. Indeed, polygonal controls belong to $H_0^1$, and the enhancement belongs to their closed signature closure almost surely. Hölder rough-path convergence at an exponent greater than $1/p$ implies $p$-variation convergence: for the level-$k$ differences, sum the bound proportional to $|t-s|^{kp\alpha/k}=|t-s|^{p\alpha}$ over a partition. Since $p\alpha>1$, these sums are uniformly bounded and tend to zero with the Hölder distance.
For the reverse inclusion, use <Brownian rough path small-ball positivity>, namely $\mathbb P(d_p(\mathbf B,e)<r)>0$ for every $r>0$, and <Cameron-Martin quasi-invariance of enhanced Brownian motion>. These are the Brownian input results used without proof here. For $h\in H_0^1$, the <rough path translation> $T_h$ is continuous and sends $e$ to $S_2(h)$. The law of $T_h\mathbf B$ is equivalent to that of $\mathbf B$. Given an open neighborhood $U$ of $S_2(h)$, continuity supplies an identity neighborhood mapped into $U$. Small-ball positivity gives positive probability that $T_h\mathbf B\in U$; equivalence of the two laws gives $\mathbb P(\mathbf B\in U)>0$. Hence $S_2(h)$ belongs to the Brownian support for every $h$. Closedness of support proves the equality with $K$.
Let $\Phi$ be the state-path solution map of the <rough differential equation> driven by these vector fields, with the fixed initial value $y_0$. The assumed identification of the Stratonovich solution says $Y=\Phi(\mathbf B)$. Boundedness of the fields and all their derivatives ensures global existence and the regularity needed by the <universal limit theorem>. That theorem makes $\Phi$ continuous from the rough-path space into the uniform state-path space. For a control signature, consistency with ordinary equations gives $\Phi(S_2(h))=y^h$.
We use the elementary <support under a continuous map> identity
$$
\operatorname{supp}(f_*\mu)=\overline{f(\operatorname{supp}\mu)}.
$$
To prove it, a neighborhood of $f(x)$ with $x\in\operatorname{supp}\mu$ has an open inverse image containing $x$, and therefore positive measure. This puts $f(\operatorname{supp}\mu)$, and its closure, in the image support. Conversely, the image closure is closed and has full image measure, since the original support has full measure. The minimal closed full-measure characterization from Question 4(i) gives the other inclusion.
Apply this identity to $f=\Phi$ and the Brownian rough-path law. We obtain
$$
\operatorname{supp}\mathcal L(Y)=\overline{\Phi(K)}.
$$
Continuity and the definition of $K$ imply $\Phi(K)\subset\overline{\{\Phi(S_2(h)):h\in H_0^1\}}$. The reverse inclusion after taking closures follows from $S_2(h)\in K$. Therefore the last displayed support is exactly the uniform closure of the skeleton paths, proving the theorem.
Finally, piecewise constant and smooth approximations of $\dot h$ converge in $L^2$, hence in $L^1$ on a finite interval. Their controls converge in total variation, their signatures converge in the rough-path metric, and the <universal limit theorem> gives uniform convergence of the corresponding skeleton solutions. This proves the stated equivalence of the control classes.
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