Four-point optimal-code conjecture for the Hilbert cube

Let $H$ be the Hilbert cube, with standard basis vectors $e_k$, and set

$$\alpha:=\frac{\sqrt{2}\pi}{3}.$$

Choose $N\subseteq\mathbb{N}$ such that

$$\sum_{k\in N}\frac{1}{k^2}=\alpha-1,$$

and define

$$s:=\sum_{k=2}^\infty\frac{1}{k}e_k,\qquad t:=\sum_{k\in N}\frac{1}{k}e_k.$$

Hilbert-cube four-point conjecture. The set

$$\left\{0,\left(1-\frac{1}{2}\alpha\right)e_1+s,e_1+t,\frac{1}{2}\alpha e_1+s-t\right\}$$

is an optimal code in the Hilbert cube. Numerical experiments on truncations motivate this prediction; the supplied text later confirms that such choices of $N$ exist but notes that the resulting codes are not unique.