Let $\{(Y_i,X_i)\}$ be i.i.d. random pairs that satisfies $$Y_i=m(X_i)+e_i, \;\; \mathsf{E}(e_i|X_i)=0, \ i\in\{1,\dotsc,n\}.$$ Let $\hat m(x)$ be an estimator of $m(x)$ at point $x$ and $\boldsymbol{X}=(X_1,\dotsc, X_n)$. If we know the conditional bias and variance converges to zero in probability: \begin{align} \operatorname{Bias}(\hat m(X)|\boldsymbol{X})&=o_p(1),\\ \operatorname{Var}(\hat m(X)|\boldsymbol{X})&=o_p(1), \end{align} then can we say something about the consistency (convergence in probability) of $\hat m(\cdot)$?
*I'm aware that if $\hat m(x)$ converges in the (unconditional) mean square sense to $m(x)$, then Markov's inequality would imply convergence in probability.
**In my case $\hat m$ is the local linear estimator.