The idea of next puzzle arises from a combination of the definition of the so-called strong primes and the Firoozbakht's conjecture. You can find the inequality defining strong primes in number theory from this Wikipedia and the Firoozbakht's conjecture from [1] or well from this Wikipedia.
On assumption of the Firoozbakht's conjecture one has that strong primes $p_{n+1}$ satisfy the inequality
$$1+\frac{p_{n+2}}{p_{n}}<2(p_n)^{1/n}.\tag{1}$$
Computational fact. Seems that there are finitely many primes $p_n$ satisfying $$1+\frac{p_{n+2}}{p_{n}}<\sqrt{3}\cdot p_n^{1/n}.$$ You can see such evidence if you this code
for (i = 1, 10000,if (1+prime(i+2)/prime(i)<sqrt(7/2)*(prime(i))^(1/i),print(prime(i))))
in Sage Cell Server (choose GP as language and press Evaluate). Notice that you can to write $3$ isntead of $7/2$ inside the square root function.
Thus using the inequalities from the literature for the size of the $n$th prime number should be easy to prove it.$\square$
My belief is that the following conjecture inspired in previous remarks is true.
Conjecture. For each fixed integer $l\geq 0$ there exist finitely many prime numbers that we denote with $P_n$ satisfying the inequality $$1+\frac{P_{n+2}}{P_{n}}<\sqrt{3+\frac{2^{l+1}-1}{2^{l+1}}}\cdot P_n^{1/n}.\tag{2}$$
Question. Can you to prove or refute previous conjecture? Many thanks.
I am asking about to prove unconditionally, or refute previous conjecture about prime numbers.
References:
[1] Farideh Firoozbakht, Conjecture 30. The Firoozbakht Conjecture, The Prime Puzzles & Problems Connection, by Carlos Rivera (22 August 2012).