Extremal persistence probabilities of exchangeable sign-invariant random variables

Let $(X_1,\ldots,X_n)$ be a random variable in $(\mathbb{R}\setminus\{0\})^n$ that is both exchangeable and sign-invariant. For every $k\in[n]$, let $S_k=\sum_{i=1}^kX_i$. Define the weak persistence probability as $\mathbb{P}(S_1,\ldots,S_n\geq0)$, and the strong persistence probability as $\mathbb