Lower-dimensional obstruction for bipartite d-degenerate graphs

Determine whether, for every integer $d\ge 3$, there exists a bipartite $d$-degenerate graph that is not pandichotomous in Euclidean dimension $d-1$.

Background

The paper proves that every $d$-degenerate graph is pandichotomous in Euclidean dimension $d$ and on the sphere of dimension $d-1$. It also constructs, for every $d\ge 2$, a $(d+2)$-degenerate bipartite graph that is not pandichotomous in Euclidean dimension $d$, while the corresponding one-degree-stronger bipartite obstruction in dimension $d-1$ remains unresolved.

The authors explicitly state their belief that the answer should be positive, but no proof is supplied, so the question qualifies as an unresolved conjectural problem.