On Wikipedia it says that the following weaker version of Grimm's conjecture is also still open:
"A product of $k$ consecutive composite numbers has $k$ pairwise distinct prime factors."
Is it true, that this is still open or is there already a proof for it?
Motivation
My motivation to this question comes from factorization theory: Let $H$ be a monoid, i.e. a commutative cancellative semigroup with identity element. An element $a \in H$ is called a strong atom, if it is an atom (i.e. irreducible) and for every $k \in \mathbb{N}$ the only facorization of $a^k \in H$ into atoms is $a\cdot...\cdot a$ (where $a$ appears $k$ times).
Let $\text{Int}(\mathbb{Z}) = \{ f \in \mathbb{Q}[x] \mid f(\mathbb{Z}) \subseteq \mathbb{Z} \}$ be the ring of integer-valued polynomials and consider the multiplicative monoid $H = \text{Int}(\mathbb{Z}) \setminus \{0 \}$ of non-zero elements.
It is known that the factorization behaviour of this monoid is very rich. For instance, for every list of integers $n_1,...,n_k \geq 2$ there exists $f \in H$ such that $f$ has exactly $k$ different (up to order and multiplication by units) factorizations into atoms and the lengths of these factorizations are exactly $n_1,...,n_k$.
The construction for the result in the preceeding paragraph only uses a very restricted pool of polynomials and does not give much information about atoms, though. That is why one is interested in a better understanding of these smallest multiplicative bricks and considers examples:
It is well-known since a long time that for $n \in \mathbb{N}$ the polynomial
$\binom{x}{n} = \frac{x(x-1)...(x-n+1)}{n!} \in H$
is an atom. These so-called binomial polynomials are of particular interest, because taken together the form a $\mathbb{Z}$-module basis of $\text{Int}(\mathbb{Z})$.
It is conjectured that the binomial polynomials are strong atoms. This is already known for several special cases. A particularly easy case is when $n$ is prime. More generally, it turns out that the distribution of prime numbers between $\max \{p \in \mathbb{P} \mid p \leq n \}$ and $n$ is crucial for the proof of the conjecture. It should be not too hard to see that it follows from Grimm's conjecture. I also have the suspicion that it implies Grimm's weak conjecture as stated above.
Thank you very much for your help!