= Solution
The <maximum-likelihood estimator> is
$$
\widehat\theta_{\mathrm{MLE}}=\overline X.
$$
Under the score-outer-product convention stated in the question,
$$
\widehat i_n
=\frac1n\sum_{j=1}^n
(X_j-\overline X)(X_j-\overline X)^T.
$$
This is the sample covariance matrix with divisor $n$. The <strong law of large numbers>, applied entrywise, gives
$$
\boxed{\widehat i_n\longrightarrow I_p\quad\text{almost surely}.}
$$
Thus the <empirical score outer-product information> consistently estimates the per-observation Fisher information.
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