= Solution
Let $T_\pi$ permute coordinates $0,\ldots,m$, fixing the later coordinates. For any integrable <random variable> $V$, its group average
$$
A_mV=\frac1{(m+1)!}\sum_{\pi\in S_{m+1}}V\circ T_\pi
$$
is invariant under this group, hence $\mathcal S_m$-measurable. If $C\in\mathcal S_m$, invariance of $C$ and measure preservation of each $T_\pi$ give $\mathbb E[\mathbf1_CV\circ T_\pi]=\mathbb E[\mathbf1_CV]$. Thus $A_mV$ satisfies the integral definition of $\mathbb E[V\mid\mathcal S_m]$. This proves <symmetrization as conditional expectation>.
Apply it to $V=\mathbf1_{A_k}(X_k)$ for $m\geq k$. Under a uniform <permutation> of $m+1$ coordinates, its selected coordinate is uniform among all of them, so
$$
\mathbb E[\mathbf1_{A_k}(X_k)\mid\mathcal S_m]
=\frac1{m+1}\sum_{j=0}^m\mathbf1_{A_k}(X_j)
=\frac{S_m^k}{m+1}.
$$
The <reverse martingale convergence theorem> says that, for decreasing sigma-algebras $\mathcal H_m$ and an integrable $V$, these conditional expectations converge <almost surely> and in $L^1$ to $\mathbb E[V\mid\bigcap_m\mathcal H_m]$. Here $\mathcal S_m\downarrow\mathcal S$, because every finitely supported permutation belongs to one of the finite groups. Therefore
$$
\boxed{\frac{S_m^k}{m+1}\longrightarrow\mathbb E[\mathbf1_{A_k}(X_k)\mid\mathcal S]\quad\text{almost surely and in }L^1.}
$$
Intersect these probability-one events for the finitely many $k=0,\ldots,p$ and multiply the limits. Expanding the product of finite sums gives
$$
\boxed{\prod_{k=0}^p\mathbb E[\mathbf1_{A_k}(X_k)\mid\mathcal S]
=\lim_{m\to\infty}\frac1{(m+1)^{p+1}}
\sum_{0\leq\ell_0,\ldots,\ell_p\leq m}\prod_{k=0}^p\mathbf1_{A_k}(X_{\ell_k})\quad\text{almost surely}.}
$$
The sum here includes repeated indices. No independence of the original coordinates was assumed; exchangeability and reverse conditional-expectation convergence are precisely what replaced it.
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