General tame $SL_k$-frieze pattern counting conjecture

Let $K=\mathbb{F}_q$ be a finite field, let $k\geq 5$ and $w\geq 1$, and set $n=w+k+1$. A tame $\operatorname{SL}_k$-frieze pattern of width $w$ over $K$ is a frieze pattern satisfying the tameness condition considered in the paper.

General counting conjecture. If $\gcd(k,w+k+1)=1$, then the number of tame $\operatorname{SL}_k$-frieze patterns of width $w$ over $K$ is

$$\frac{(q^n-q)(q^n-q^2)\dots(q^n-q^{k-1})}{\lvert \operatorname{PGL}(k,K)\rvert}.$$

This formula extends the corresponding results for $k=2,3,4$ and suggests a uniform count in the coprime case, but it remains conjectural for $k\geq 5$.