Solution
= Solution
Weighted least squares uses the known conditional variances and should be at least as efficient, so $\rho\leq1$. Indeed, apply Cauchy--Schwarz to
$$
\frac{|X_1|}{\sqrt{v(X_1)}}\quad\text{and}\quad |X_1|\sqrt{v(X_1)}.
$$
It gives
$$
\bigl(\mathbb E[X_1^2]\bigr)^2
\leq\mathbb E[X_1^2/v(X_1)]\,\mathbb E[v(X_1)X_1^2],
$$
which is exactly $\rho\leq1$. Equality holds precisely when $v(X_1)$ is constant on the part of the support where $X_1\ne0$, up to null sets.