Suppose we have two estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ of $\theta$, both with the same bias.
If we have
$$ \begin{align} &\hat{\theta}_1 \xrightarrow{a.s.}\ \theta \\ &\hat{\theta}_2 \nrightarrow_{a.s}\ \theta \text{ but }\hat{\theta}_2 \xrightarrow{p}\ \theta \end{align} $$
Question: Do we necessarily have
$$ Avar(\hat{\theta}_1) \le Avar(\hat{\theta}_2)?$$
Here, $Avar$ denotes the asymptotic variance.
That is, is a strongly consistent always (asymptotically) preferable to one that is known not to be strongly consistent?