Claim in OEIS will give non-trivial results about Legendre's and Brocard's conjecture. The claim is very likely to be true, but I am not sure it is currently provable.

Brocard's conjecture states that $\pi({p_{n + 1}}^2) - \pi({p_n}^2) > 3$ for all $n > 1$.

Legendre's conjecture states that there is always prime between $n^2$ and $(n+1)^2$.

Relation between some OEIS sequences:

A108309 Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row. Except for the initial term, a(n)>=2 because in the interval 2n-1 of odd numbers there are always at least two primes. For n>2, this is the same as the number of primes between n^2-n and n^2+n, which is the sum of A089610 and A094189. - T. D. Noe, Sep 16 2008

Question: Is it true that $A108309(n) \ge 2$ except for the initial term?

The prime gap can be as large as $C \log{x}$ infinitely often and the claim doesn't appear to address this at all.

Number of primes between n^2 and (n+1/2)^2. Same as the number of primes between n^2 and n^2+n. Oppermann conjectured in 1882 that a(n)>0. - T. D. Noe, Sep 16 2008

Number of primes between n^2-n and n^2 (inclusive)

So these comments imply there are at least two primes between $n^2-n$ and $n^2+n$. The size of this interval $2n+1$ is essentially the same as the interval in Legendre's conjecture, but the starting point is not square.

Since $n^2-n > (n-1)^2$ and $n^2+n < (n+1)^2$, there are at least two primes between $(n-1)^2$ and $(n+1)^2$, which is closely related to both conjectures.