Regularity-independence conjecture for higher secondary polytopes

Let $\mathcal{A}$ be a point configuration, let $\mathcal{V}$ be its associated vector configuration, and let $\Upsigma_{\mathcal{A},k}$ be the higher secondary polytope defined using fine regular zonotopal tilings of $\mathcal{Z}_{\mathcal{V}}$. For a fine zonotopal tiling $T$, write $\widehat{\operatorname{vert}}_k(T)$ for its associated vector. The regularity-independence conjecture.

$$\Upsigma_{\mathcal{A},k}=\operatorname{conv}\left\{\widehat{\operatorname{vert}}_k(T)\;\middle|\; T\text{ is a fine zonotopal tiling of }\mathcal{Z}_{\mathcal{V}}\right\}.$$

Equivalently, $\widehat{\operatorname{vert}}_k(T)$ lies in $\Upsigma_{\mathcal{A},k}$ for every fine, not necessarily regular, zonotopal tiling $T$. The conjecture would show that regularity can be omitted from the definition of the higher secondary polytope; the supplied source gives no resolution.