Monotonicity conjecture for a gamma-function quotient

Let $a,c\in\mathbb{R}$ satisfy $0\le a\le 1$ and $c>4$. Consider the quotient

$$\frac{\Gamma(c)\Gamma(c-4)}{\Gamma(c-a)\Gamma(c+a-1)}.$$

Gamma-quotient monotonicity conjecture. The quotient decreases as $c$ increases.

The conjecture extends the preceding lemma, which establishes the same monotonicity for $1<a<4$ and $-3<a<0$. The source notes that the remaining range $0\le a\le 1$ is suggested by computer experimentation, but provides no proof or resolution.