Intermediate Borel asymptotic separation index

Determine whether there exists a locally finite Borel graph whose Borel asymptotic separation index is strictly greater than one but finite.

Background

The paper defines the Borel asymptotic separation index as a parameter measuring the number of smooth or finite-class pieces needed to control bounded neighborhoods in a Borel graph. It proves that finite Borel asymptotic dimension implies asymptotic separation index at most one. The authors leave unresolved whether finite values strictly between one and infinity can occur.