Existence of regular $3$-hypertopes with $2^n$ chambers

For any positive integers $n,s,t,l$ such that $n\ge 10$, $s,t\ge 2$, $l\ge 1$ and $n\ge s+t+l$, a new infinite family of regular 3-hypertopes with type $(2^s,2^t,2^l)$ and automorphism group of order $2^n$ is constructed.