Engbers–Galvin conjecture on colorings of the Hamming cube

Let $Q_d$ be the $d$-dimensional Hamming cube, let $N=2^d$ be its number of vertices, and let $C_q(Q_d)$ denote the number of proper $q$-colorings of $Q_d$. For fixed $q$, define

$$f(q)=\frac{\lceil q/2\rceil}{2\lfloor q/2\rfloor}\left(2-\frac{2}{\lceil q/2\rceil}\right)^d+\frac{\lfloor q/2\rfloor}{2\lceil q/2\rceil}\left(2-\frac{2}{\lfloor q/2\rfloor}\right)^d.$$

Engbers–Galvin conjecture. For each fixed $q$,

$$C_q(Q_d)=(1+\mathbf{1}_{\{q\text{ odd}\}})\binom{q}{\lfloor q/2\rfloor}(\lfloor q/2\rfloor\lceil q/2\rceil)^{N/2}\exp\bigl[(1+o(1))f(q)\bigr]$$

as $d\to\infty$. The first factors count the choices of a main phase and the pure colorings with that phase, while the exponential accounts for isolated flaws. The conjecture is stated as the general fixed-$q$ prediction, although the paper notes that for $q>4$ it can be weaker than the actual asymptotics.