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Claim. Theorem 1 of S. Laishram and T. N. Shorey, Grimm's conjecture on consecutive integers, Int. J. Number Theory 2 (2006), no. 2, 207--211: Grimm's conjecture holds for $n\leq p_{N_0}$ and for all $k$, where $N_0=8.5\times10^8$ and $p_{N_0}=19236701629$. That is, whenever $n\leq p_{N_0}$ and $n+1,\ldots,n+k$ are all composite, there are distinct primes $P_i\mid n+i$ for $1\leq i\leq k$. Theorem 2 reduces this to the maximal runs of composites between consecutive primes, $n=p_N$ and $k=p_{N+1}-p_N-1$ for $1<N\leq N_0$. Its proof combines Philip Hall's theorem on distinct representatives with a Sylvester–Erdős argument and a Mathematica computation (card). The journal gives the June 2006 issue and no day, so the page is dated by the first of that month.

Covers. Every run of composites starting after $n\leq19236701629$, of any length, for which the problem's distinct primes exist. Larger $n$ are not covered, so the problem stays open.

Acceptance. The result appeared in a refereed journal, the International Journal of Number Theory, in 2006: the refereed evidence. The site labels the problem FALSIFIABLE, an open label, so its commentary's credit is not reviewed evidence.