Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\Omega^1_X\to\cdots\to\Omega^{p-1}_X.$$
For $p = 1,2$ we have the well known quasi-isomorphisms:
$$\mathbf{Z}(1)_{\mathcal{D}}\simeq\mathbf{G}_m[-1],\ \ \mathbf{Z}(2)_{\mathcal{D}} = [\mathcal{O}_X^{\times}\xrightarrow{\rm dlog}\Omega^1_X][-1].$$
We also have a multiplicative structure on $\mathbf{Z}(p)_{\mathcal{D}}$ under cup products.
Is the map
$$\mathbf{Z}(1)_{\mathcal{D}}^{\otimes 2}\to\mathbf{Z}(2)_{\mathcal{D}}$$
a quasi-isomorphism?
In higher weights it sure isn't. Any explicit example, if not?