McKean conjectured that Gaussian random variables are optimal for the $n$-th order derivative of differential entropy along the heat flow, and verified this for $n=1$, $2$. Recently, Zhang, Anantharam and Geng introduced the linear matrix inequality approach to show that this conjecture holds for $n\leq 5$ under the log-concavity assumption. In this work, with the same assumption, we improve their method using the positive semidefinite reformulation and validate McKean's conjecture for $n\leq 9$, and also the completely monotone conjecture for $n\leq 11$.