Every $A \in \text{GL}_n(\mathbb{R})$ has a unique [Polar decomposition][1]: 

$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$.
In particular the orthogonal factor is given by $$O_A=A(\sqrt{A^TA})^{-1}.$$ **Question:**

Let $A,B \in \operatorname{GL}_n^+$. Does there exist a positive constant $c<1$ such that $$ |AB-O_{AB}| \ge c|AB-O_AO_B|$$

(Where the norm $| \cdot |$ is the standard Euclidean/Frobenius norm)


**Motivation:**

$O_A$ is the [closest matrix in $\text{SO}_n$ to $A$][2]; We can think of it as the isometric projection of $A$. This projection is not multiplicative in general** (i.e $O_{AB} \neq O_AO_B $ for some $A,B$, for a concrete example see [here][3]).  

My question concerns the boundedness of the "error" (in computing the distacne from $\text{SO}_n$) when using $O_A  O_B$ instead of $O_{AB}$.
_____

** Indeed, see [this previous question][4] of mine, which concerns the analysis of the pairs of matrices satisfying the multiplicativity property.

[1]:https://en.wikipedia.org/wiki/Polar_decomposition
[2]:https://math.stackexchange.com/questions/1477855/distance-of-a-matrix-from-the-orthogonal-group
[3]:https://math.stackexchange.com/a/1910835/104576
[4]:http://mathoverflow.net/questions/248842/when-does-isometric-projection-respect-multiplication?rq=1