Schmutz Schaller's counting-function conjecture for square and hexagonal lattices

For $j\geq 1$, let $b_j(n)=1$ if $n$ is represented by $X^2+jY^2$, and $b_j(n)=0$ otherwise. Define

$$B_i(x)=\sum_{n\leq x}b_i(n),\qquad i\in\{1,3\}.$$

Schmutz Schaller's counting-function conjecture. For every $x$, one has

$$B_1(x)\geq B_3(x).$$

This is an equivalent reformulation of the square-versus-hexagonal lattice conjecture. The paper proves the inequality for every $x$, so the conjecture is solved.