Turn point of a non planar tensioned rope 

Let a rope be fixed at a point "A". The other end of the rope (point "B") is attached (tangentially) to a horizontal drum which is 2m higher than point "A". The drum stretches the rope.
A vertical pillar is located off the vertical plane defined by points A & B. The rope is redirected by the pillar - i.e. it goes from "A" to the pillar and then turns to point "B".
At which height along the pillar will the tensioned rope turn?

My reflections are that the rope either follows a steady slope, or the height along  the pillar is the projection of the slope defined by points A & B, or that the length of the rope should be minimum. However I cannot find any source regarding this matter, and I cannot reach a proof for my reflections.
 A: At mechanical equilibrium the potential energy has to be at the minimum, see, e.g., https://en.wikipedia.org/wiki/Minimum_total_potential_energy_principle, and for any reasonable rope material the potential energy of the stretched rope should be a monotonically increasing function of its length. Thus the length of the rope between points A and B is minimized by the proper choice of y. This can be solved formally by finding the minimum of the length of the rope given by
$L(y)=(l_1^2+y^2)^{1/2} + (l_2^2+(H-y)^2)^{1/2}$,
where $l_1$ and $l_2$ are horizontal projections of distance from points A and B to the bending point, and H is the height of point B.  
Writing $dL/dy=0$ immediately leads to the relation $\frac{y}{l_1} = \frac{H-y}{l_2}$ which is of course the statement of constant slope. The advantage of solving it formally as a minimization problem is that one can apply this principle to similar but more difficult problems, with multiple drums and pillars that have finite diameters, non-trivial cross-section shapes etc.
A: If the rope can slide freely up or down the pillar, then its slope wrt the ground will be constant - ie the same between A and C as between C and B. This is because if we "unwind" the rope and there is a "kink" in it where the slope changes, then there is a component of tension which pulls the rope straight. If there is nothing to oppose this lateral force (such as friction with the pillar) then the rope will slide up/down until it becomes straight, to minimise tension. 
After "unwinding" the rope from the pillar so that it is straight, by similar triangles we have :
height at C / height at B = distance AC / distance AB.
If there is friction between the rope and the pillar it will prevent the rope from becoming "straight." The height of the rope on the pillar is then difficult to predict, and depends where it was placed initially.
