The all-dimensional fundamental-group injectivity conjecture for Rips-to-shadow projections

Let $X$ be a finite subset of Euclidean space, let $\mathcal{R}(X)$ be its Rips complex, and let $\mathcal{S}(X)$ be its shadow complex. Let

$$p\colon\mathcal{R}(X)\to\mathcal{S}(X)$$

be the projection. The all-dimensional fundamental-group injectivity conjecture. The projection $p$ induces an injection on fundamental groups in all dimensions. The authors motivate this by stating that they believe $\pi_1$-injectivity holds in general; no resolution is given in the source.