Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$?

Mientka and Weitzenkamp [MiWe69] proved that the only $n \leq 2^{44}$ such that $n > 2$ and $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$ are $4, 7, 15, 21, 45, 75, 105$.