Open. The site labels the problem OPEN and credits no solution. The standing derives from the claim pages: the two pending partial claims are Martins 2026, a manuscript that bounds the primes and the block length of any sufficiently large solution, proves finiteness for each fixed $k$ and block length, and treats the case $k=2$ in detail, and Mysore 2026, a draft note showing that any further solution with $k=2$ is a product of at least $1924$ consecutive primes; neither decides whether solutions are infinite, so the problem is open with no settling or pending full claim.