Assuming the Lenstra–Pomerance–Wagstaff conjecture, in what range of values for $p$ would we expect to find the next prime $2^p - 1$?

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Assuming the Lenstra–Pomerance–Wagstaff conjecture is correct, in what range of values for a prime $p$ would we expect to find the next Mersenne prime $2^p - 1$?