Almost-diameter conjecture for affine and projective actions

Let $n\boxed{\geq 2}$ and $q\geq 1$. Define

$$A_q=\{(x_1,\ldots,x_n)\in(\mathbb{Z}/q\mathbb{Z})^n:\gcd(q,x_1,\ldots,x_n)=1\}$$

as the primitive vectors, and let

$$P_q=A_q/(\mathbb{Z}/q\mathbb{Z})^\times.$$

The group $\mathrm{SL}_n(\mathbb{Z})$ acts naturally on both spaces. For $x,y$ in either space, define

$$d(x,y):=\min\{\log\|\gamma\|:\gamma\in\mathrm{SL}_n(\mathbb{Z}),\ \gamma\cdot x=y\}.$$

Almost-diameter conjecture. For every $\epsilon>0$, and for $q$ large enough depending on $\epsilon$, the following holds.

1. For every $(x,y)\in A_q^2$ outside a set of exceptions of size $\epsilon|A_q|^2$, it holds that

$$d(x,y)\leq \left(\frac{1}{n-1}+\epsilon\right)\log q.$$

2. For every $(x,y)\in P_q^2$ outside a set of exceptions of size $\epsilon|P_q|^2$, it holds that

$$d(x,y)\leq \left(\frac{1}{n}+\epsilon\right)\log q.$$

This is an optimal almost-diameter statement for the natural affine and projective actions, and is related to the optimal lifting property. The conjecture is known for $P_q$ when $q$ is prime and $n=3$, while several partial results are available; the general case remains open.