Universal odd-prime valuation bound for the sum of divisors

Let $p>2$ be a fixed prime, and let $n\in\mathbb{N}$. Write $\sigma(n)$ for the sum of the positive divisors of $n$, and let $\nu_p$ denote the $p$-adic valuation. For a prime $q\mid n$, write $\nu_q(n)$ for the exponent of $q$ in $n$. The relevant conditions are: (1) every prime $q$ dividing $n$ satisfies $\nu_p(\sigma(q^{\nu_q(n)}))\leq\lfloor\log_p(q^{\nu_q(n)})\rfloor$; or (2) every prime $q$ dividing $n$ that satisfies $\nu_p(\sigma(q^{\nu_q(n)}))=\lceil\log_p(q^{\nu_q(n)})\rceil$ is less than $p$.

Universal odd-prime valuation conjecture. Every $n\in\mathbb{N}$ satisfies condition (1) or (2). Consequently,

$$u_p(\sigma(n))\leq\lceil\log_p n\rceil.$$

A complete characterization of the indices $n$ for which equality holds in this bound remains open, although the bound is proved under the stated conditions.