A sharp monotonicity and upper-bound conjecture for binomial tail ratios

Let $B_{n,k}(p)$ be the binomial tail probability, $b_{n,k}(p)$ the corresponding binomial point probability, and $L(n,k,p)$ the lower bound from the paper's Theorem~. Suppose $k,n\in\mathbb{N}_0$, $k\leq n$, and $0<p<1$. Tail-ratio conjecture. The quantity

$$\frac{B_{n,k}(p)}{b_{n,k}(p)L(n,k,p)}$$

is nondecreasing in $k$ and nonincreasing in $n$ and $p$. In addition,

$$\frac{B_{n,k}(p)}{b_{n,k}(p)}<\frac{180451625}{143327232}L(n,k,p)\qquad\forall k\leq pn,$$

and

$$\frac{B_{n,k}(p)}{b_{n,k}(p)}<\sqrt{\frac{\pi}{2}}L(n,k,p)\qquad\forall k\leq pn-1.$$

The conjecture seeks a sharp description of the ratio between the binomial tail and point probabilities, improving the factor-two bounds established in the paper; the stated numerical constant is approximately $1.25902$, while the conjecture is proved only in a special case using a connection with Ramanujan's equation.