The reciprocal 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

$$\mu_f(x)=\sum_{n\leq x}\frac{f(n)}{n}.$$

Reciprocal counting-function conjecture. For every $x$, one has

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

The paper notes that the main counting-function conjecture implies this one, but the supplied text does not state a separate proof or resolution.