Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d−k)/2$, where the parameters k,d are integers that satisfy $0 \lt k \lt d$. What is the best possible upper bound for $\mathbb{P}\{X \lt k/d\}$?

Edit: This problem has a beautiful geometric interpretation as the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-dimensional subspace.

So far, after several days of toil, the best result I have been able to prove is a wretched 0.922. This is based on a nontrivial normal approximation proposed by Alfers & Dinges (1984) and refined by Artstein (2002).

Numerical examples suggest that the extremal case occurs when $k=1$ for every $d \geq 1$. Furthermore, for this case, the limiting value as $d\to\infty$ appears to be equal to $\Phi(1)-\Phi(−1)$, where $\Phi$ is the standard normal cdf.

I can obtain considerably better bounds for this special case, but I have no idea how to prove that these parameters are extremal. Any thoughts on how to solve this problem?