If $\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\geq 1$,

$$\liminf_{n\to \infty}\sum_{0\leq i<k}\omega_k(n+i)\leq k?$$

Is it true that

$$\limsup_{n\to \infty}\left(\sum_{0\leq i<k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1,$$

where $\omega$ counts the number of distinct prime factors without restriction?