The arrival conjecture for Collatz-invariant measures

Let $\mathrm{Col}:\mathbb N\to\mathbb N$ be the Collatz map, and let $\mu$ be a finitely additive $\mathrm{Col}$-invariant measure on $\mathbb N$, with every singleton measurable. Define its arrival sequence by $a_n:=\mu(\{n\})$ for $n>0$. Arrival conjecture. For all $k\geq 1$ such that $a_k>0$, we have $a_1>0$.

The conjecture is presented as equivalent to the Collatz conjecture and expresses that positive mass at any state forces positive mass at $1$. The source gives no resolution status for this formulation, so it remains open here.