Let $(X,d)$ be a finite metric space. Let $C$ be a Hamiltonian cycle (over $X$) outputted by Christofide's algorithm. Also, let $K$ be a minimum spanning tree. I am aware that Christofide's algorithm is a $\frac{3}{2}$ approximation algorithm. But does the following also holds
$$l(C) \le \frac{3}{2} l(K)$$
where $l(.)$ denotes the cost, interpreted in a natural way. I do not see why it should hold, but this paper seems to use this as a fact in theorem $5$.