Concavity conjecture for normalized Bell numbers

For $x>0$, define

$$B(x)=\frac{1}{e}\sum_{k=0}^{\infty}\frac{k^x}{k!},$$

so that $B(n)=B_n$ for nonnegative integers $n$, where $B_n$ is the $n$-th Bell number. Bell-function concavity conjecture. The function $B(x)^{1/x}$ is log-concave for $x\geq1$, equivalently

$$(\log B(x)^{1/x})”<0\qquad (x>1).$$

This would imply Sun's conjectured log-concavity of $\{\sqrt[n]{B_n}\}_{n\geq1}$. The paper proves that $B(x)^{1/x}$ is strictly increasing, but gives no resolution of the stronger concavity conjecture.