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

$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$. 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 $A$; 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).

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 of mine, which concerns the analysis of the pairs of matrices satisfying the multiplicativity property.