On heights of distributivity matrices

We construct a model in which there exists a distributivity matrix of regular height $\lambda$ larger than $\mathfrak{h}$; both $\lambda = \mathfrak{c}$ and $\lambda < \mathfrak{c}$ are possible. A distributivity matrix is a refining system of mad families without common refinement. Of particular interest in our proof is the preservation of $\mathcal{B}$-Canjarness.