The extension-monad conjecture for rings and their additive reducts

Let $R$ be a ring and let its additive reduct be the algebra obtained by retaining the additive operation and forgetting multiplication. Write the corresponding extension monads for these two structures as extension monads of $R$ and of its additive reduct.

Extension-monad conjecture. The extension monad of a ring has the same underlying set as the extension monad of its additive reduct.

This conjectural property is motivated by the behavior of the extension monad of the ring of integers and is suggested as a possible principle extending to dibinary distributive algebras. The supplied text does not state whether the conjecture has been resolved.