The conjecture that a gamma-function expression is logarithmically completely monotonic

Let $\psi$ denote the digamma function, let $\gamma$ denote the Euler–Mascheroni constant, and define

$$q(t):=t^{t(\psi(t)-\log t)-\gamma}.$$

Logarithmic complete monotonicity conjecture. The function $q$ is logarithmically completely monotonic on $(0,\infty)$; that is,

$$(-1)^n[\log q(t)]^{(n)}\geq 0$$

for every $t>0$ and every integer $n\geq 1$. This conjecture concerns the relationship between completely monotonic and logarithmically completely monotonic functions, with the latter forming a subclass of the former. The supplied text attributes the conjecture to an earlier work but gives no evidence of a resolution.