Upper bounds for the monotone rank of the unique disjointness matrix

It is shown that the $\mathsf{O}\mathsf{R}$-rank (covering rank) of the $2^n\times 2^n$ unique disjointness matrix is $n^{O(1)}(3\mathrm{/}2{)}^n$, hence the known lower bound ${1.5}^n$ turns out to be essentially tight. By the way, an upper bound ${1.89}^n$ is obtained for the $\mathsf{S}\mathsf{U}\mathsf{M}$-rank (partition rank) of this matrix.