I had asked this question in math.stackexchange (link: https://math.stackexchange.com/questions/1902276/bounds-on-the-moore-penrose-inverse-of-a-product ) but I did not get any response so I am trying my luck here.

Let $A^{\dagger}$ denote the Moore-Penrose inverse of a real matrix and let $\|A\|$ denote the usual matrix norm given by the largest singular value of $A.$

Then is it true that $\|(AB)^{\dagger}\| \leq \|A^{\dagger}\| \|B^{\dagger}\|?$

This is trivially true whenever $(AB)^{\dagger} = B^{\dagger}A^{\dagger}$, which happens, for example, when $A$ is full column rank and $B$ is full row rank. But what about the general case?