Equality of the union and indexed annihilator families

Let $V$ be a Banach algebra, and let $X$ be a Banach space. Let $\varphi\!:V\to X$ be a bounded linear map. For $z\in V$, write ${\mathfrak{C}}_{-1}(z)$ for the set of corresponding good-path coefficients, and let ${\mathfrak{S}}_\cup^\varphi(z)$ and ${\mathfrak{S}}_c^\varphi(z)$ denote the associated families. Equality conjecture. For all $z\in V$,

$${\mathfrak{S}}_\cup^\varphi(z)=\bigcup_{c\in{\mathfrak{C}}_{-1}(z)}{\mathfrak{S}}_c^\varphi(z).$$

The preceding corollary establishes the inclusion from right to left; the conjecture asserts that every element of ${\mathfrak{S}}_\cup^\varphi(z)$ arises from one coefficient $c\in{\mathfrak{C}}_{-1}(z)$.