Given matrices $A,B \in \mathbb{R}^{m,n}$, the orthogonal Procrustes problem asks to find an orthogonal matrix $Q \in \mathbb{R}^{n \times n}$ such that
$$\|AQ - B\|_F^2$$
is minimized. There is a simple closed-form solution to this problem.
I am interested in a variant of this problem where we would like to find both an orthogonal matrix $Q$ as before and a diagonal matrix $\Lambda \in \mathbb{R}^{m \times m}$ such that
$$\|\Lambda AQ - B\|_F^2$$
is minimized.
Is this a known problem with a closed-form solution? It is basically matching two sets of points where a single rotation on one set and individual scaling on each point in that set is allowed.