On the probability that a discrete IFR random variable exceeds its mean

Let $X$ be an integer-valued random variable with $\mathbb{E}(|X|)<\infty$ whose survival probabilities $S_j=\mathbb{P}(X\ge j), j\in\mathbb{Z}$, form a log-concave sequence. Suppose that for some integer $m$ and some $γ\in (0, 1/2]$ it holds $\mathbb{E}(X)\ge m$ and $\mathbb{P}(X=m+1)\geγ\cdot\math