Let $p_t$ and $q_t$ be two families of probability densities on $\mathbb{R}^d$ indexed by time $t\geq 0$. Does the following differential inequality imply that the KL-divergence is identically zero?
$$\frac{d}{dt}\text{KL}(p_t,q_t) \leq \int_\mathbb{R}^d (\nabla \log q(x))^T \left(\Sigma_q - \Sigma_p \right) (\nabla \log p_t(x) - \nabla \log q_t(x)) p_t(x) dx, \quad \text{KL}(p_0,q_0)=0$$
The matrices $\Sigma_p$ and $\Sigma_q$ are the covariance matrices of $p$ and $q$.
If $p$ and $q$ are Gaussians with mean $0$, the inequality does imply that $\text{KL}(p_t,q_t) = 0$, I think.