For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since 
$$6=1+1+4=1+2+3=2+2+2.$$


**QUESTION**. Is it true that the inequality 
$$p(n,k)^2>p(n,k-1)p(n,k+1)\tag{1}$$
holds whenever $1\le k\le n-26$?

I conjecture that the answer is positive, this implies that for any integer $n\ge27$ the integer sequence $(p(n,k))_{1\le k\le n-26}$ is log-concave. For $n=27,28,\ldots,100$, I have verified the
inequality $(1)$ for all $k=1,\ldots,n-26$.

Your comments are welcome! If you know that the question is not new, please provide me a reference.