Optimal-colouring conjecture for fixed-length de Bruijn cycle decompositions

Let $\mathcal{E}(q,k,\ell)$ denote the maximum number of eBugs in an $\ell$-valid colouring with $q$ colours and $k$ LEDs per eBug. An optimal colouring is one attaining the upper bound $\mathcal{E}(q,k,\ell)\leq q^\ell/k$; necessarily $k>\ell$. Optimal-colouring conjecture.

$$\mathcal{E}(q,k,\ell)=\dfrac{q^\ell}{k}$$

whenever $k$ divides $q^\ell$ and $k>\ell$. This characterises when optimal colourings exist and would resolve the exact determination problem in these divisibility cases; the conjecture was confirmed computationally for all $q,\ell$ with $q^\ell\leq81$, and is known for $\ell=2$ by Bryant's decomposition result.