Infinite family of cubic counterexample graphs

Determine whether there exist infinitely many 3-edge-colourable cubic graphs \(G\) such that for every pair of edges \((u,v),(x,y)\in E(G)\), \(\alpha(G)=\alpha(G-\{u,v,x,y\})\).

Background

The paper constructs the Biggs–Smith graph as a 3-edge-colourable cubic graph with the property that deleting the endpoints of any two edges does not reduce its independence number. A computational search also identifies a second example, the 168-vertex graph F168D.

The authors explicitly ask whether these examples belong to an infinite family of 3-edge-colourable cubic graphs having the same independence-number invariance property.