Best-constant Stolarsky-mean approximation conjecture for the elliptic integral

Let $r^{\prime}=\sqrt{1-r^{2}}$. For $p\neq 0$, let $S_{p,q}(a,b)$ denote the Stolarsky mean, and let $p_{0}\approx 1.763135$ be the unique solution on $(0,9/4]$ of

$$\left(\frac{9/2-p}{p}\right)^{1/(2p-9/2)}=\frac{2}{\pi},$$

that is, $\theta_{p_{0}}=2/\pi$. The best-constant Stolarsky-mean conjecture. There exists $r_{0}\in(0,1)$ such that

$$H(r)=\frac{1-(2/\pi)E(r)}{1-S_{9/2-p_{0},p_{0}}(1,r^{\prime})}$$

is strictly increasing on $(0,r_{0})$ and strictly decreasing on $(r_{0},1)$. Consequently,

$$\frac{2}{\pi}E(r)<S_{9/2-p_{0},p_{0}}(1,r^{\prime})$$

for $r\in(0,1)$, with the best constant $p_{0}$. The paper proposes this claim as a conjecture, and the supplied text gives no resolution evidence.