Averaging Two Noisy Measurements

Again the structural model is $Y=2X+u$ with $E[u\mid X]=0$, but now you observe two noisy proxies: $$W_1=X+\eta_1, \qquad W_2=X+\eta_2,$$ where $\eta_1,\eta_2$ are independent of each other and of $X,u$. Suppose $\operatorname{Var}(X)=4$ and each noise term has variance 1.

If you regress $Y$ on the average proxy $\bar W=(W_1+W_2)/2$, what is the probability limit of the slope?