Growth conjecture for partial sums 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 $(\mathbb N,\leq_{\mathcal T})$. Growth conjecture. There is a positive constant $C$ such that

$$\sum_{i=1}^n\mu_{\mathcal T}(i)\leq-Cn$$

for all sufficiently large $n$. This predicts a negative linear trend for the partial sums, unlike the corresponding classical Möbius-function sums; it is presented as an empirically motivated conjecture and remains unresolved.