InverseSeries giving incorrect result

Somehow in Mathematica 13.2.0.0, InverSeries generates incorrect results.

Let's look at the following two series that differs from each other by a constant number "$1$"

InverseSeries[Series[1 + 1/(2 g) + 2/3 (1/g)^(3/2), {g, \[Infinity], 3}], x] InverseSeries[Series[1/(2 g) + 2/3 (1/g)^(3/2), {g, \[Infinity], 3}], x] 

Both outputs are $$ \frac{1}{2 x}+\frac{2 \sqrt{2}}{3 \sqrt{x}}-\frac{8}{9}+\frac{40 \sqrt{2} \sqrt{x}}{27}+O\left(x^1\right)$$

However, this is only correct for the second input, the first output should be $$ \frac{1}{2 (x-1)}+\frac{2 \sqrt{2}}{3 \sqrt{(x-1)}}-\frac{8}{9}+\frac{40 \sqrt{2} \sqrt{(x-1)}}{27}+O\left((x-1)^1\right)$$