Cruickshank–Garamvölgyi–Jordán–Nixon conjecture on rigidity with added loops

Let $G=(V,E,L)$ be a looped simple graph. For a positive integer $k$, let $G^{[k]}$ be the graph obtained by adding $k$ new loops at every vertex, and call $G$ $k$-tight if every vertex subset $X\subseteq V$ induces at most $k|X|$ edges and loops and $|E\cup L|=k|V|$. Cruickshank–Garamvölgyi–Jordán–Nixon conjecture. If $d$ and $t$ are positive integers with $d\geq 2t$, then $G^{[d-t]}$ can be realised as an infinitesimally rigid linearly constrained framework in $\mathbb{R}^d$ if and only if $G$ has a $t$-tight looped simple spanning subgraph. This would extend the known characterisation of rigidity for sufficiently looped graphs to the general range covered by $d\geq 2t$; the source presents it as an open conjecture.