Conjecture on short intervals containing integers with many distinct prime factors

Let $\omega(n)$ denote the number of distinct prime factors and define

$$\mathcal{A}=\left\{n:\omega(n)\geq\frac{C_0\log_2 n}{\log_3 n}\right\},$$

where $\log_2 n=\log\log n$ and $\log_3 n=\log\log\log n$. Short-interval prime-factor conjecture. For some $C_0\geq1$, there is a constant $1\leq d<C_0$ such that, for sufficiently large $x>0$,

$$\mathcal{A}\cap\left(x-\left(\log\frac{x}{2}\right)^d,x\right]\neq\varnothing.$$

The source presents this as a weaker version sufficient to disprove Erdős Problem #679. Its resolution is not given.