In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{Post}(\mathcal{C})$ of "Postnikov pretowers" in $\mathcal{C}$: towers of the form $\cdots \to X_2 \to X_1 \to X_0$ in $\mathcal{C}$ in which each map $X_{n+1} \to X_n$ exhibits $X_n$ as the $n$-truncation of $X_{n+1}$. By HTT Proposition 5.5.6.26, "Postnikov towers in $\mathcal{C}$ are convergent" if and only if both conditions below hold. (The labels of these conditions are based on the way this proposition is formulated in HTT.)
(1) => (2): Every object $X \in \mathcal{C}$ is the limit of its Postnikov tower.
(2) => (1): Every Postnikov pretower $\cdots \to X_2 \to X_1 \to X_0$ is the Postnikov tower of its limit $X = \lim X_n$.
In Spectral Algebraic Geometry, Lurie changes the terminology to "$\mathcal{C}$ is Postnikov complete" (Warning A.7.2.2). This is sensible both because $\mathrm{Post}(\mathcal{C})$ is a kind of completion of $\mathcal{C}$, and also because "Postnikov towers in $\mathcal{C}$ are convergent" might reasonably be understood to refer to the condition (1) => (2) alone.
My question is: Does anyone know an example of a presentable $\infty$-category (ideally, an $\infty$-topos) which satisfies (1) => (2), i.e. "every object is the limit of its Postnikov tower", but not (2) => (1), and therefore is not Postnikov complete? Or is it possible that condition (1) => (2) actually implies Postnikov completeness?
