Will Cramer Rao Lower Bound converge to the variance of MLE in the below probability distribution?

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The pdf is given such as,
$f(x;\theta) = (\frac{1}{\sqrt{2\theta\pi x^3}}e^{\frac{-(x-1)^2}{2\theta x}})$

Is it necessary that in this case, the Maximum Likelihood Estimator will be equal to the CRLB in this case?