Thank you Prof. Johnson. May I get a suitable reference which covers these topics in detail? Or if time permits, can you kindly sketch how to choose specific $T_i$'s so that the product equals the infimum?

Oh. I see. Thank you for the explanation. So basically it is equivalent to the some mentioned gap between consecutive primes. You may close the question if you wish.

May be it's now of historical value, but still it's unsolved. I was discussing with my friend on its implications. On the wiki, we just found that its truth allows to have stricter bound on gaps of primes (as mentioned at the wikipedia page). So I was looking if there is some literature on this.