The completely monotone conjecture for Fisher information along the heat flow

Let $X$ be a random variable such that $t\longmapsto I(X,t)$ is infinitely differentiable.

Completely monotone conjecture. The function $t\longmapsto I(X,t)$ is completely monotonous, that is

$$(-1)^n\partial_t^n I(X,t)\geq 0$$

for every $t>0$ and non-negative integer $n$.

The conjecture asks whether all derivatives of Fisher information along the heat flow have alternating signs. The first cases are known, but the statement is not known for all $n$; the surrounding discussion specifically notes that the cases $k\geq 4$ remain unresolved.