Dris's biconditional conjecture for odd perfect numbers

Let $N=q^k n^2$ be an odd perfect number in Eulerian form, where $q$ is prime, $q\equiv k\equiv 1\pmod 4$, and $\gcd(q,n)=1$. Let $\sigma(m)$ denote the sum of the positive divisors of $m$, and let $\nu_q(N)$ denote the exponent of $q$ in $N$. Dris's biconditional conjecture. The following biconditional holds:

$$k=\nu_q(N)=1\Longleftrightarrow \sigma(n)<q^k.$$

This conjecture links Sorli's conjecture to an inequality involving the divisor sum of the non-Euler component. It is presented as a consequence suggested by the paper's new results and remains open.