Average absolute value conjecture for the triangular-number Möbius function

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the Möbius function of the poset $(\mathbb N,\leq_{\mathcal T})$. Average absolute value conjecture. As $n\to\infty$,

$$\sum_{i=1}^n|\mu_{\mathcal T}(i)|=\frac12n+o(n).$$

Equivalently, the average of $|\mu_{\mathcal T}(i)|$ tends to $\frac12$. This is based on numerical data and is not resolved in the supplied text.