Russell's conjecture for level 2 modular equations

Let $p$ be an odd prime, and write

$$\frac{p+1}{4}=\frac{n}{d},$$

where $n$ and $d$ are respectively the numerator and denominator after simplification. Define $u$ and $v$ by

$$u^h=\alpha\beta,\qquad v^h=(1-\alpha)(1-\beta),$$

with $h=4/d$.

Russell's conjecture for level 2. There exists a polynomial $P(u,v)$ of degree $(p+1)/2$ in $u$ and $v$ such that

$$P(u,v)=0.$$

This is the level-2 analogue of the stated Russell and Chan–Liaw modular-equation theorems, and it is intended to support the modular-polynomial procedure used to derive Ramanujan-type series for $1/\pi$. Its resolution is not established by the supplied text.