Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows, $ M_{A, B} = 1 $ if $A \subseteq B$; $M_{A, B} = -1$ if $B \subsetneq A$; $M_{A, B}$ can take arbitrary value over $\mathbb{R}$.

In words, $M$ is a matrix with some missing entries. Is there some result lower bounds the rank of matrix $M$, for example, $rk(M) \ge n^{\Omega(\log n)}$.