Polynomial 0-embracing exchange distance for simplices

Establish whether any two 0-embracing simplices in dimension d can be transformed into one another by a 0-embracing exchange sequence of length at most d+1.

Background

The paper studies the problem of transforming one simplex whose convex hull contains the origin into another by exchanging vertices one at a time, while requiring every intermediate simplex to contain the origin in its convex hull. The minimum number of exchanges is called the 0-embracing exchange distance.

Computational experiments by Caoduro, Khodamoradi, Shepherd, and Paat motivated the conjecture that this distance is always at most d+1. The paper explicitly notes that the conjecture remains unresolved.