Polynomials $f$ and $f'$ with all roots distinct integers

The arXiv paper posted here (pdf) contains a list of references that have broached this problem in the past, and examples for $n=3$, $4$, and $5$ (which I have since incorporated into the OP).

However, even the case of $n = 6$ is listed as open$^\star$ (cf. Open Problem 6 on PDF 23/24) as of 2004. So, it appears the question asked here is open.

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$\star$ (Edit): User i9Fn helpfully points to a 2015 paper containing the sextic polynomial

$$f(x) = (x − 3130)(x + 3130)(x − 3590)(x + 3590)(x − 10322)(x + 10322)$$

which leads to

$$f'(x) = 6x(x − 3366)(x + 3366)(x − 8650)(x + 8650)$$

thereby resolving the above open problem, and leading to an updated open question (cf. p. 363):

Are there polynomials with the above-stated features and degree greater than six?

According to this latter paper's author, no such examples are known.