Conjecture on short intervals containing integers with many prime factors

For $x>0$, define

$$\mathcal{A}=\{n:\omega(n)\geq\varepsilon\log_2 n\},\qquad \mathcal{B}=\{n:\Omega(n)\geq\varepsilon\log_2 n\},$$

where $\log_2 n=\log\log n$. Short-interval conjecture. For $\varepsilon>0$, there is a constant $C$ such that, for sufficiently large $x$,

$$\mathcal{A}\cap\bigl(x-C\log x\sqrt{\log_2 x},x\bigr]\neq\varnothing,$$

and

$$\mathcal{B}\cap\bigl(x-C\log x\sqrt{\log_2 x},x\bigr]\neq\varnothing.$$

This conjecture is motivated by density estimates for integers with a prescribed number of prime factors; the source says a weaker version would suffice for its disproof of Erdős Problem #679. Its resolution is not given.