The 3-dimensional shadow inclusion connectivity conjecture

Let $X\subseteq\mathbb{R}^3$ be finite and let $v\in X$. Write $X_v$ for the subset associated with the vertex $v$, $X^v$ for the corresponding closed neighborhood, and $X-v$ for $X\setminus\{v\}$. Let $\mathcal{S}$ denote the shadow complex, and let a map be $1$-connected when it induces an isomorphism on $\pi_0$ and a surjection on $\pi_1$. The 3-dimensional shadow inclusion connectivity conjecture. The inclusion map

$$\mathcal{S}(X_v)\hookrightarrow\mathcal{S}(X^v)\cap\mathcal{S}(X-v)$$

is $1$-connected. This is presented as a strengthening of an earlier proposition; the paper notes that it would imply that the projection from the Rips complex to the shadow complex is $2$-connected for finite subsets of $\mathbb{R}^3$, and hence induces a $\pi_1$-isomorphism. Its resolution is not specified in the source.