Matthews' conjectures for relatively prime generalized Collatz maps

Let $d\geq2$, let $m_{0},\ldots,m_{d-1}$ be nonzero integers, and choose $r_j\in\mathbb{Z}$ with $r_j\equiv jm_j\pmod d$. Define $T_{\mathrm{Matthews}}:\mathbb{Z}\to\mathbb{Z}$ by

$$T_{\mathrm{Matthews}}(x)=\frac{m_jx-r_j}{d}\quad\text{when }x\equiv j\pmod d.$$

Call the map relatively prime when $\gcd(m_j,d)=1$ for every $j$. Matthews' conjectures. In the relatively prime case: (I) every trajectory is eventually periodic if $\prod_{j=0}^{d-1}|m_j|<d^d$; (II) if $\prod_{j=0}^{d-1}|m_j|>d^d$, the union of divergent orbit classes has density $1$ in $\mathbb{Z}$; (III) regardless of these inequalities, the map has a finite, nonzero number of cycles; and (IV) every point in a divergent orbit class has iterates uniformly distributed modulo $d^n$ for every $n\geq1$, meaning

$$\lim_{N\to\infty}\frac{1}{N}\left|\left\{k\in\{0,\ldots,N-1\}:T^{\circ k}(x)\equiv j\pmod{d^n}\right\}\right|=\frac{1}{d^n}.$$

These are stated as a package of conjectures about the generalized map and remain open in the source.