A note on Noether symmetries and conformal Killing vectors

A conjecture was stated in Hussain et al. (Gen Relativ Grav 41:2399, 2009), that the conformal Killing vectors form a subalgebra of the symmetries of the Lagrangian that minimizes arc length, for any spacetime. Here, a counter example is constructed to demonstrate that the above statement is not true in general for spacetimes of non-zero curvature.