The blow-up conjecture for globally rigid graph constructions

Let $G$ be a connected graph with $v>1$ vertices, denoted $v_1,\ldots,v_v$. Replacing each vertex $v_i$ by an independent set of size $a_i$ and connecting the new independent sets according to the edges of $G$ gives a graph constructed from $G$ by independent-set blow-ups.

Blow-up conjecture. For any connected graph $G$ with $v > 1$, there exists some $a_1, a_2, \ldots, a_v$ and some $d$ such that if we replace each $v_i$ with an independent set of size $a_i$ and connect them accordingly, the resulting graph is GPR in $\mathbb{R}^d$.

This conjecture proposes a globally rigid construction for every connected graph with more than one vertex, extending the paper's experimentally observed and proved examples for irreducible graphs of small order, $k$-chains, and $k$-rings. The general assertion remains open in the supplied text.