Künneth conjecture for Čech homology of product cosheaves

Let $M$ be a manifold, let $l$ be a positive integer, and let $k_1,\ldots,k_l$ be degrees. For a cover $\mathcal{U}$ of $M^l$, suppose that every intersection of elements of $\mathcal{U}$ is diffeomorphic to a finite union of Euclidean spaces. Denote by $Z^{k_1} \mathbin{\widehat{\otimes}} \dots \mathbin{\widehat{\otimes}} Z^{k_l}$ the corresponding product cosheaf on $M^l$. Künneth conjecture.

$$\check{H}_\bullet(\mathcal{U}, Z^{k_1} \mathbin{\widehat{\otimes}} \dots \mathbin{\widehat{\otimes}} Z^{k_l}) \cong \check{H}_\bullet(M, Z^{k_1}) \mathbin{\widehat{\otimes}} \dots \mathbin{\widehat{\otimes}} \check{H}_\bullet(M, Z^{k_l}).$$

This is the expected Künneth formula for the topological Čech homology of the product cosheaves. The preceding discussion explains that product covers are not generally cofinal among covers of $M^l$, so the formula cannot be obtained directly from the projective-limit description; its validity is left open in the paper.