Medians are below joins in semimodular lattices of breadth 2

Let $L$ be a lattice of finite length and let $d$ denote the minimum path length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in L^k$,anelement$y$belongingto$L$iscalledamedianof$\xi$ifthesum$d(y,x_1)+\cdots+d(y,x_k)$isminimum\mathrm{.}Thelattice$L$satisfiesthe$c_1$-medianpropertyif,forany$\xi=(x_1,\dots,x_k)\in L^k$ and for any median $y$ of $\xi$, $y\le x_1\vee \cdots \vee x_k$. Our main theorem asserts that if $L$ is an upper semimodular lattice of finite length and the breadth of $L$ is less than or equal to $2$, then $L$ satisfies the $c_1$-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that $2$ cannot be replaced by $3$.