Johnson–Leader–Russell's tower-of-cubes conjecture for maximal chains

Let $X$ be an $n$-element set, and let $\binom{X}{\bullet}$ denote the Boolean lattice of subsets of $X$. For integers $t,k\geq 1$ with $n=tk$, partition $X$ into pairwise disjoint blocks $X_1,\dots,X_k$ of size $t$. The tower of $t$-cubes is the family

$$\mathcal{T}_t=\left\{A\subseteq X:(X_1\cup\cdots\cup X_s)\subseteq A\subseteq(X_1\cup\cdots\cup X_{s+1})\text{ for some }0\leq s\leq k-1\right\}.$$

For a family $\mathcal{A}\subseteq 2^X$, let $c(\mathcal{A})$ be its number of maximal chains. Johnson–Leader–Russell's tower-of-cubes conjecture. If $|\mathcal{A}|=|\mathcal{T}_t|$, then

$$c(\mathcal{A})\leq c(\mathcal{T}_t).$$

The conjecture asserts that towers of cubes maximize the number of maximal chains among equally sized families in the Boolean lattice. The source states that these constructions are equivalent, from the set-system perspective, to bucket orders, and explicitly says that the conjecture is disproved by improved constructions based on regular bipartite posets.