Noise sensitivity of bootstrap percolation for more than two neighbours

Let $\mathcal{C}_n$ denote the event of complete occupation for $k$-neighbour bootstrap percolation on the sequence of tori $\mathbb{T}_n^d$ and boxes $[n]^d$.

Noise sensitivity conjecture. For any $d greater than or equal to k > 2$, the event $\mathcal{C}_n$ is noise sensitive.

The result is known for $d greater than or equal to k=2$, but for $d greater than or equal to k>2$ the equivalence between complete occupation and the existence of a suitable simple seed is not known, leaving the mechanism governing pivotal sites unclear.