Propagation-time interval conjecture for minimum zero forcing sets of hypercubes

Let $Q_d$ be the $d$-dimensional hypercube graph. A minimum zero forcing set is a zero forcing set of cardinality equal to the zero forcing number of $Q_d$; its propagation time is the number of parallel forcing rounds needed to color all vertices. Let $\operatorname{pt}(Q_d)$ and $\operatorname{PT}(Q_d)$ denote the minimum and maximum propagation times, respectively, among minimum zero forcing sets of $Q_d$. A graph has a full propagation time interval if it has minimum zero forcing sets with every integer propagation time between $\operatorname{pt}(Q_d)$ and $\operatorname{PT}(Q_d)$.

Propagation-time interval conjecture. For every $d\geq 2$,

$$\operatorname{pt}(Q_d)=1 \quad\text{and}\quad \operatorname{PT}(Q_d)=2^{d-2}.$$

Moreover, $Q_d$ has a full propagation time interval.

The preceding results establish the existence of minimum zero forcing sets with propagation times $1$, $2$, $3$, and $4$ for $d\geq 4$, and hence already give non-automorphic minimum zero forcing sets. The conjecture predicts the complete range of propagation times through the claimed maximum $2^{d-2}$; the source states that this remains open, together with the enumeration of all minimum zero forcing sets of $Q_d$.