Fix two positive integers $n,m$ such that $n>2m$ and $n\ge 3$, and let $X\sim \text{Bin}(n,\frac{m}{n})$ be a binomial random variable. For each $i\in \{1,\ldots,m\}$, set $\alpha_i = \mathbb{P}(X\le m-i) - \mathbb{P}(X\ge m+i)$. It is known that $\alpha_1>0$, a result that is referred to as Simmons' inequality.
Computer simulations suggest that the following refinement of Simmons' inequality may hold true:
Conjecture: For $k\in\{1,\ldots,m\}$, let $S_k := \sum_{i=1}^{k} \alpha_i$. Then it holds $S_k \ge 0$, for all $k\in\{1,\ldots,m\}$.
Hence $S_1\ge 0$, by Simmons' inequality. Moreover, it holds $S_m\ge 0$. To see this, note that $$ m = \sum_{i=1}^{m} \mathbb{P}(X\ge i) + \sum_{i=m+1}^{2m} \mathbb{P}(X\ge i) + \sum_{i\ge 2m+1} \mathbb{P}(X\ge i) $$
and transfer the first two sums on the right-hand side to the other side to deduce $$ S_m = \sum_{i\ge 2m+1} \mathbb{P}(X\ge i) \ge 0.$$
I wonder if there are known results for the remaining values of $S_k$. Any help would be very much appreciated.
