The limiting shift conjecture for the Gamma-function expansion

Let $(v_n)_{n\geq 0}$ be the sequence of shifts appearing in the asymptotic expansion

$$\sum_{n \geq 0} \frac{g_n}{(x+v_n)^{2n}}.$$

The sequence begins with $v_1=\frac{23}{90}$, $v_2=\frac{1792627}{7289730}$, and $v_3=\frac{570984637359867601981}{2288928529497568067550}$; the term $v_0$ is irrelevant because it occurs with zero power. Limiting shift conjecture.

$$\lim_{n\to\infty}v_n=\frac14.$$

The conjecture is motivated by the numerical values of the first several shifts, which approach $1/4$, the constant shift occurring in a related expansion of the Gamma function. No proof or disproof is supplied in the source.