About the minimum of the Gamma function on $(0,1)$

Question

Problem :

Denotes by $x_{min}=k$ the minimum (not the abscissa) of the Gamma function $x!$ on $(0,1)$ then prove or disprove that :

$$\left(e^{-\frac{k^{2}}{C^2}}\right)!>k$$

Where $C=-1+\frac{1}{\ln(3)}+\ln(3)$

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I cannot find an attempt because the problem of the minimum of $x!$ is really hard . In consequence I have tried numerical tools and it seems true for the first five digit ($0.8856$)

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Question :

How to (dis)prove it ? Have you seen this equation before in the litterature ?

Thanks !