Generalized Legendre Conjecture: Is there a prime between ns and (n+1)s for s < 2?

Equipped with the functions `isPrime(n)` and `nextPrime(n)`, we can now easily test hypotheses and conjectures about primes. (A conjecture is a statement we believe to be true but have not proved; when someone eventually proves a conjecture, it becomes a theorem.) Here are some interesting conjectures – all of them apparently true: [Click to show or hide discussion.] Legendre's conjecture.    For each integer n > N2 = 0, there is a prime p between n2 and (n+1)2. The n5/3 conjecture.    For each integer n > N5/3 = 0, there is a prime p between n5/3 and (n+1)5/3. The n8/5 conjecture.    For each integer n > N8/5 = 0, there is a prime p between n8/5 and (n+1)8/5. The n3/2 conjecture.    For each integer n > N3/2 = 1051, there is a prime p between n3/2 and (n+1)3/2. The n4/3 conjecture.    For each integer n > N4/3 = 6776941, there is a prime p between n4/3 and (n+1)4/3. The n5/4 conjecture.    For each integer n > N5/4 ≥ 50904310155, there is a prime p between n5/4 and (n+1)5/4. The n6/5 conjecture.    For each integer n > N6/5 ≥ 833954771945899, there is a prime p between n6/5 and (n+1)6/5.

The above statements are verifiable up to at least 18-digit primes. (The existing knowledge of maximum prime gaps up to low 19-digit numbers simplifies the verification; beyond that, the verification becomes less practical.) These statements suggest the following generalization: The generalized Legendre conjecture.
(A) There exist infinitely many pairs (s, Ns), 1 < s ≤ 2, such that for each integer n > Ns there is a prime between ns and (n+1)s. (Weak formulation.) (B) For each s > 1, there exists an integer Ns such that for each integer n > Ns there is a prime between ns and (n+1)s. (Strong formulation.)
Discussion.  Some of the pairs (s, Ns) are the above special cases: (2,0), (5/3, 0), (8/5, 0), (3/2, 1051), (4/3, 6776941). Ns is a function of s; namely, Ns denotes the greatest counterexample for the ns conjecture: if we proceed from small to large n, the last interval [ns, (n+1)s] containing no primes happens to occur at n = Ns; and for each n > Ns there is at least one prime between ns and (n+1)s. We readily see that s > 1 is indeed a necessary condition. (A short explanation for a younger reader: should we have s ≤ 1, our intervals [ns, (n+1)s] would be way too narrow to contain a prime – or any integer at all, in most cases). Moreover, is it plausible that s > smin > 1 should also be satisfied, with a certain lower bound smin> 1, in order for Ns to exist? If so, part (A) would still be true, but part (B) would be invalidated for some s very close to 1.

Questions to explore further:
(1) Is Ns a monotonic function of s? (No, it is not – there are counterexamples.)
(2) What the lower bound smin might be, for the ns conjecture to still make sense? (A tongue-in-cheek guess: less than 1.1. A more serious answer: s > 1 is likely enough; no additional smin is needed. Here is a hint.)
(3) How is the ns conjecture related to other known conjectures and theorems about the distribution of primes? (The ns conjecture follows from the Cramer-Granville conjecture when n is large enough.)

Here is a partial computational verification of special cases of the generalized Legendre conjecture.