Local characterization of the cap family by cup families

Let $V$ be a Banach algebra, let $Y$ be a Banach space, and let $\varphi\!:V\to Y$ be a bounded linear map. For $z\in V$, let ${\mathfrak{S}}_\cap^\varphi(z)$ and ${\mathfrak{S}}_\cup^\varphi(z)$ be the associated families. Local characterization conjecture. For every $z\in V$,

$${\mathfrak{S}}_\cap^\varphi(z)=\bigcup\left\{\bigcap_{z'\in U}{\mathfrak{S}}_\cup^\varphi(z'):\ U\subseteq V\text{ is open and bounded, }z\in U\right\}.$$

The conjecture seeks to recover the cap family at $z$ from cup families on open bounded neighborhoods of $z$.