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on Tableaux problem 48

The statement

48.[FaigHerr1981a] via [Far] shows that a finite modular lattice $L$ with join-irreducible elements $J$ is characterized by the partial ordering induced on $J$ and a function $\rho:J\times J\rightarrow 2^{J}:(a,b)\mapsto\{c\in J\,|\,c\leq a\vee b\}$ ; if two modular lattices have isomorphic posets $J$ and functions $\rho$ , they are isomorphic. Can this be extended to a classification by describing necessary and sufficient criteria for $\rho$ for it to correspond to a modular lattice?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.