Conjecture on the weighted sum of 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 $(\mathbb N,\leq_{\mathcal T})$. Weighted-sum conjecture. There is a positive constant $E$ such that

$$\lim_{n\to\infty}\sum_{i=1}^n\frac{\mu_{\mathcal T}(i)}{i}=-E.$$

This appears in the paper's section of further questions as a proposed asymptotic statistic for $\mu_{\mathcal T}$; the supplied text gives no evidence of a proof or disproof.