Unbounded magnitude 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 $(\mathbb N,\leq_{\mathcal T})$. Unbounded magnitude conjecture. For any positive integer $M$, there is a positive integer $n$ such that

$$|\mu_{\mathcal T}(n)|\geq M.$$

This formalizes the empirical observation that the triangular-number Möbius values attain arbitrarily large magnitude, in contrast with the classical Möbius function; no resolution is supplied.