Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is *locally finite*: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \bigoplus_{n \in \mathbb{Z}} P_n$ is *graded-projective* if there is a graded free module $F = \bigoplus A(l_i)$ (for some shifting parameters $l_i \in \mathbb{Z}$) and a graded module $Q$ such that $F \cong P \oplus Q$ as graded modules. We say that $P$ is *bounded below* if $P_n = 0$ for $n \ll 0$.

In section 2 of *The structure of AS Gorenstein algebras* by Minamoto and Mori, it is shown that if $P$ is a left $A$-module that is graded, projective (as an "ungraded" module), and bounded below, then it is graded-projective. Nakayama's lemma seems crucial to their methods, so I expect something to go wrong if $P$ is not bounded below.

Question:Is there a locally finite graded algebra $A$ and a graded left $A$-module $P$ that is projective, not bounded below, and not graded-projective?

If something indeed does go wrong, I would already expect it to occur in the case where $A$ is *connected*, meaning that $A_0 = k$. (This question arose earlier on Math StackExchange.)