Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.)

Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\epsilon \log\log m$?

Barreto and Leeham, using ChatGPT and Aristotle, have proved a negative answer, which was quantified by Tao and Alexeev (see the comments).