  The "Modularity Conjecture" is the assertion that the join of two nonmodular
varieties is nonmodular. We establish the veracity of this conjecture for the
case of linear idempotent varieties. We also establish analogous results
concerning $n$-permutability for some $n$, and the satisfaction of nontrivial
congruence identities. Our theorems require a technical result about the
equational theory of linear varieties, which might be of independent interest.
