Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number Theorem to prove that $$\frac{1}{x}\sum_{n \in S:\, n \le x} = \frac{\log^{k-1} \log x}{(k-1)!\log x} (1+o(1)),$$ and he obtains some information on the size of the error term.

I am trying to find what is known, conditionally (on RH) and unconditionally, on the asymptotics of $\sum_{n\in S: x\le n \le x+x^{c}} 1$ as $x$ goes to infinity. Landau's result answers this for $c=1$.

For $k=1$, I am just asking about actual primes in short intervals. RH gives the asymptotics when $c>1/2$, and the work of Huxley gives the asymptotics unconditionally when $c>7/12$.
For $k>1$ all I can find are 'almost everywhere' results. What results are known for *all* intervals when $k>1$?