A construction of A. Schinzel — many numbers in a short interval without small prime factors

Hardy and Littlewood (1923) conjectured that for any integers $x,y\ge2$
\begin{equation} \label{HL} \pi(x+y) \le \pi(x) + \pi(y). \end{equation}

Let us call a set $\{b_1,\dots,b_k\}$ of integers admissible if for each prime $p$ there is some congruence class $\bmod p$ which contains none of the integers $b_i$. The prime $k$-tuple conjecture states that if a set $\{b_1,\dots,b_k\}$ is admissible, then there exist infinitely many integers $n$ for which all the numbers $n+b_1,\dots,n+b_k$ are primes.
Let $x$ be a positive integer and $\rho^*(x)$ be the maximum number of integers in any interval $(y,y+x]$ (with no restriction on $y$) which are relatively prime to all positive integers $\le x$. The prime $k$-tuple conjecture implies that
$$\max_{y\ge x}(\pi(x+y)-\pi(y))=\limsup_{y\ge x} (\pi(x+y)-\pi(y))=\rho^*(x).$$

Hensley and Richards (1974) proved that
$$\rho^*(x) - \pi(x) \ge(\log 2- o(1)) x(\log x)^{-2}\quad(x\to\infty).$$
Therefore, (\ref{HL}) is not compatible with the prime $k$-tuple conjecture. Using a construction of Schinzel we show that
$$\rho^*(x) - \pi(x) \ge((1/2)- o(1)) x(\log x)^{-2}\log\log\log x\quad(x\to\infty).$$