Idempotent profiles in graph products of monoids

For a monoid, consider the equivalence relation defined by equality of idempotent left identities. We give a partial answer to whether its left-congruence condition is preserved by ordinary monoid graph products: it is preserved when every vertex monoid is left Fountain. The proof transports local replacements through arbitrary left contexts using published first-block and replacement results. A separate embedding theorem shows that unital vertex embeddings preserving this equivalence and introducing no new one-sided invertible elements induce equivalence-preserving graph-product embeddings. As a corollary, graph products of finite monoids for which the equivalence equals R* satisfy the left-congruence condition. The unrestricted question for arbitrary vertices is not settled. The principal results retain candidate status.