C*-extreme implies linear extreme for UCP on arbitrary Hilbert spaces

Determine whether, for arbitrary Hilbert spaces H and unital C*-algebras A, every C*-extreme point of the C*-convex set UCP(A,B(H)) is a linear extreme point of UCP(A,B(H)).

Background

Proposition 2.4 shows that when H is finite-dimensional, C*-extreme points of UCP(A,B(H)) are linear extreme points. The authors explicitly state that it is unknown whether this inclusion continues to hold when H is an arbitrary Hilbert space, highlighting a gap between finite- and infinite-dimensional settings.