Quaternionic gate-net covering conjecture for $PSU(2)$

Let $t\in\mathbb{N}^*$, let $S^3\subset\mathbb{R}^4$ be the unit $3$-sphere, and let $\nu(5^t)$ denote the integer solutions of

$$x_1^2+x_2^2+x_3^2+x_4^2=5^t.$$

For $x\in\nu(5^t)\cup\nu(5^{t-1})$, write $x/\sqrt{|x|}$ for the corresponding normalized point, and let $B_G(c,r)$ be the ball of radius $r$ centered at $c$ for the metric $d_G$. Quaternionic gate-net covering conjecture. Given $t\in\mathbb{N}^*$, $S^3$ can be covered by balls centered at each $x/\sqrt{|x|}\in\widehat{\nu}(5^t)\cup\widehat{\nu}(5^{t-1})$ with radius $r=t^n\cdot 5^{-t/4}$, where $n$ is a fixed constant:

$$S^3\subset\bigcup_{x\in\nu(5^t)\cup\nu(5^{t-1})}B_G\left(\frac{x}{\sqrt{|x|}},r\right).$$

This weaker covering statement is presented as sufficient to prove the upper bound $K(T)\leqslant 4/3$ for the typical-case compilation exponent. Its resolution is not supplied in the source.