Equidistribution conjecture for the signs of square-free representation differences

For each odd square-free integer $t$, let $\Delta(t)=r_{Q_1}(t)-r_{Q_2}(t)$, and define

$$N_+(X)=\#\{t\leq X:t\text{ is odd and square-free},\ \Delta(t)>0\},$$ $$N_-(X)=\#\{t\leq X:t\text{ is odd and square-free},\ \Delta(t)<0\},$$

and $N_{\ne 0}(X)=N_+(X)+N_-(X)$. Sign equidistribution conjecture. As $X\to\infty$,

$$\frac{N_+(X)}{N_{\ne 0}(X)}\longrightarrow\frac12,\qquad \frac{N_-(X)}{N_{\ne 0}(X)}\longrightarrow\frac12.$$

Equivalently, $N_+(X)\sim N_-(X)$. The numerical data suggest that the two signs of the difference of the representation numbers occur equally often among odd square-free integers for which the difference is nonzero; the preceding theorem gives quantitative lower bounds for both signs, but does not establish this equidistribution.