  In this paper we study indecomposable rank 2 modules in the Grassmannian
cluster category ${\rm CM}(B_{5,10})$. This is the smallest wild case
containing modules whose profile layers are $5$-interlacing. We construct all
rank 2 indecomposable modules with filtration $\{i,i+2,i+4,i+6,i+8\}\mid
\{i+1,i+3,i+5,i+7,i+9\}$, classify them up to isomorphism, and parameterize all
infinite families of non-isomorphic rank 2 modules.
