Let $R$ be a ring with 1, $M$ be a left $R$-module. Then $M$ is fp-*injective* if every $R$-homomorphism from a finitely presented left ideal to $M$ extends to a homomorphism of $R$ to $M$ i.e. if $\operatorname{Ext}^1(R/I, M) = 0$ for every finitely presented left ideal $I$. And $M$ is fp-*projective* if $\operatorname{Ext}^1(M, N) =0$ for any fp-injective left $R$-module $N$.
It is known that $R$ is left Noetherian iff every left $R$-module is fp-projective. It is known also that if $R$ is left Noetherian, then every fp-injective left $R$-module is injective.

My conjecture is that if $R$ is a left Noetherian, and every cyclic left $R$-module embeds in projective, then $R$ is a left self-injective. I was trying solve this question by using the concept of fp-projectivity, but I didn't solve it. Could someone give the answer?