Aryan–Madhavani–Parikh–Sengupta–Zhang finiteness conjecture for 2-near perfect numbers

Let $n=2^k p^m$, where $p$ is an odd prime, and suppose that $n$ is 2-near perfect when

$$\sigma(n)=2n+d_1+d_2$$

for distinct positive divisors $d_1,d_2$ of $n$. Finiteness conjecture. There are only finitely many 2-near perfect numbers of the form $2^k p^m$ for $m\geq 3$. The paper states that its main results disprove this conjecture, conditional on the standard conjecture that infinitely many Mersenne primes exist; the conditional disproof is explained by the later characterization theorem.