Fibonacci multiplier conjecture for permuted van der Corput sequences

Let $p_n=\mathfrak{F}_n$ be the $n$th Fibonacci number and let $\mu_n=\sigma_{p_{n-1},0}$. Define $z(1)=z(2)=z(3)=1$, $z(4)=2$, and, for subsequent indices, $z(n)=z(n-1)+z(n-3)+z(n-4)$. Fibonacci multiplier conjecture. The maximum of the discrepancy function satisfies

$$\max_{x\in[0,1]}\psi_{p_n}^{\mu_n}(x)=\psi_{p_n}^{\mu_n}\left(\frac{z(n-2)}{p_n}\right)=\psi_{p_n}^{\mu_n}\left(\frac{z(n-1)}{p_n}\right).$$

Moreover, the dominant interval is

$$\left[\frac{z(n-2)}{p_n},\frac{z(n-2)+1}{p_n}\right],$$

and

$$\alpha_{p_n}^{\mu_n}=\lim_{m\to\infty}\frac{1}{m}\sum_{j=1}^m\psi_{p_n}^{\mu_n}(\hat{x}_n b^j),\qquad \hat{x}_n=\sum_{m=1}^{\infty}\frac{z(n-2)}{b^m}.$$

These Fibonacci-based affine permutations are motivated by the continued-fraction behavior of consecutive Fibonacci ratios and the expected optimality of the golden-ratio rotation. The assertions are numerical observations presented as a conjecture; no general proof is given.