He and Ye's free-vertex conjecture for 4-regular 4-uniform hypergraphs

Let $H$ be a 4-regular 4-uniform hypergraph, meaning that every edge has four vertices and every vertex belongs to four edges. A free vertex is a vertex $v$ such that $V(H)\setminus\{v\}$ can be $2$-colored so that every edge contains vertices of both colors, with $v$ uncolored.

He and Ye's free-vertex conjecture. Every $4$-regular $4$-uniform hypergraph contains a free vertex.

This is the $k=4$ case of the broader free-set conjecture, but it was posed separately because the case $k=4$ is more difficult than the cases $k\in\{5,6,7,8\}$. The supplied text does not report a resolution.