Anosov homotopy of bi-contact doubling flows

Determine whether the partially hyperbolic projectively Anosov flows produced by the bi-contact doubling construction can be homotoped, through partially hyperbolic projectively Anosov flows, to an Anosov flow.

Background

The bi-contact doubling construction produces projectively Anosov flows on oriented doubles of compact bi-contact plugs. When the original plug is attracting and partially hyperbolic, the doubled flow is nonsingular and partially hyperbolic, with invariant tori separating the two copies and carrying Morse–Smale dynamics.

The paper explicitly conjectures that the answer to the displayed homotopy question is negative. Thus the unresolved issue is whether these doubled partially hyperbolic projectively Anosov flows lie in an Anosov homotopy class within the corresponding partially hyperbolic projectively Anosov category.