By playing around with assoc. Legendre polynomials, I arrived at

$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$

Now, I want to show that we don't have equality for $x \in (-1,1).$

I undertook quite some computations in order to be sure that this is really the case, but I currently don't see why this is true.

There are some simple ways to start with:

For $x=0$ the inequality is obvious and both sides are even functions.