A dyadic-denominator conjecture for lonely runner instances

Let $\mathbf{n}$ be a speed vector with $\gcd(\mathbf{n})=1$, and let a suitable time mean a time at which the speeds in $\mathbf{n}$ form a lonely runner instance. Let $k$ and $n_1$ be the parameters appearing in the speed vector, let $m$ be a natural number, and let $\lceil\cdot\rceil$ denote the ceiling function.

Dyadic-denominator conjecture. For every $\mathbf{n}$ with $\gcd(\mathbf{n})=1$, there is always a suitable time of the form

$$\frac{m}{2^{\lceil\ln_{2}(n_1)+1\rceil}(k+1)n_1}$$

for some natural number $m$.

The claim proposes an explicit restricted family of denominators containing a suitable time for every primitive speed vector. It is presented as a continuation of results giving explicit suitable times for known families; its general validity is open.