Minimum singular value of sum of rotations

Consider orthogonal matrices $$Q_1, Q_2 \in \mathbb{R}^{d \times d}$$, where both matrices are proper rotations of angle $$\theta_1, \theta_2$$ around different axes. Now, consider the symmetric matrix

$$A = \frac{Q_1 Q_2 + Q_2^\top Q_1^\top}{2}$$

Question: is there a nontrivial lower bound for $$\sigma_{\min}(A)$$ in terms of $$\theta_1, \theta_2$$?

If $$Q_1$$ was the trivial rotation, we would have $$\sigma_{\min}(A) \geq \cos(\theta_1)$$. However, it is unclear to me what happens after composing the two rotations.