Infinitude conjecture for near-perfect numbers with a prescribed redundant divisor

A positive integer $n$ is near-perfect if it is the sum of all its proper divisors except one proper divisor, called its redundant divisor. For a fixed integer $k\geq 1$, the redundant divisor is $2^k$. Infinitude conjecture. For every $k\geq 1$, there exist infinitely many near-perfect numbers with redundant divisor $2^k$. The theorem preceding this conjecture gives one Euclid-like family, but the assertion that infinitely many such numbers exist for every $k$ remains open.