The Boolean-algebra subfamily extremal conjecture

Let $m=2^n$, and consider families of $m$ sets. A subfamily forms a Boolean algebra of dimension $d$ if it is isomorphic to the Boolean lattice $2^{[d]}$. Boolean-algebra subfamily extremal conjecture. The family containing the largest number of subfamilies forming a Boolean algebra of dimension $d$ is $2^{[n]}$. This proposes that the full Boolean lattice is extremal for the number of $d$-dimensional Boolean-algebra subfamilies among families of $2^n$ sets; the source gives no resolution status or further conditions.