Universality of weighted-sum persistence for finite-variance increments

Let $(X_i)_{i\ge1}$ be i.i.d. with centered law $\mu$, let $(\sigma_i)_{i\ge1}$ be deterministic, define

$$\Sigma_n=\sum_{i=1}^n\sigma_iX_i,\qquad T_\sigma=\inf\{n\ge1:\Sigma_n>0\}.$$

Weighted-sum persistence conjecture. If $\mu$ has finite variance and $\sigma_n\asymp n^p$ for some $p>0$, then

$${\mathbb{P}}[T_\sigma>n]=n^{-(p+1/2)+o(1)}.$$

This removes the exponential-moment assumption from the preceding theorem and is posed as a natural question.