The weighted counting-function conjecture for square and hexagonal lattices

Let $b_1$ and $b_3$ be the characteristic functions of the integers represented by $X^2+Y^2$ and $X^2+3Y^2$, respectively. Define

$$\lambda_f(x)=\sum_{n\leq x}f(n)\log n.$$

Weighted counting-function conjecture. For $x\geq 8$, one has

$$\lambda_{b_1}(x)\geq\lambda_{b_3}(x).$$

The paper presents this as a related conjecture after proving the unweighted inequality, but the supplied text gives no resolution of it.