I searched a twin prime of the form $$k\cdot 2023!\pm1$$ with positive integer $k$ as a project related to the current year. User hardmath claimed to have checked the range upto $k=2\ 200\ 000$ with no result. I accidently found out that $$k=2\ 377\ 271$$ gives a twin-prime pair, not with brute force , but with random trials.
Is this the smallest $k$ giving the desired twin-prime pair ?
According to hardmath , this is the only $k$ upto $2.8\cdot 10^6$. I suggested in a chat to answer here , but so far without a reply. So, apart from a doublecheck (which is still appreciated) , this question is solved. I keep waiting for an answer.