Newton series conjecture for the principal inverse gamma function

Let $\tilde{\Gamma}(x)$ denote the principal branch of the inverse of the gamma function, so that $\tilde{\Gamma}(x)=y$ when $Gamma(y)=x$. Let $alpha=1.4616\ldots$ be the unique positive number satisfying $psi(\alpha)=0$. Newton series conjecture. The inverse gamma function has the representation

$$\tilde{\Gamma}(x)=2+\sum_{n=0}^{\infty}\sum_{k=0}^{n}k\left(\left(\prod_{i=1}^{k}\ln\left(\frac{\left(k+1\right)!}{i!}\right)\right)\prod_{i=1}^{n-k}\ln\left(\frac{\left(k+1\right)!}{\left(k+1+i\right)!}\right)\right)^{-1}\prod_{i=1}^{n}\ln\left(\frac{x}{i!}\right).$$

The series is conjectured to converge on the interval $(\Gamma(\alpha),\infty)$. This would provide a Newton-series representation of the principal inverse gamma function on the stated interval; the parser supplies no evidence resolving the convergence claim.