Bucher–Machacek's red-size conjecture for connected quivers

Let $Q$ be a connected quiver on $n$ vertices that does not admit a reddening sequence. The red size $\operatorname{Red}(Q)$ is the maximal number of red vertices obtained after mutating only at green vertices, and the unrestricted red size $\operatorname{uRed}(Q)$ is the maximal number of red vertices obtained after an arbitrary mutation sequence. A vertex is the last remaining green vertex after a sequence if it is the only green vertex left at the end of that sequence. Bucher–Machacek's red-size conjecture. The following statements hold:

1. $\operatorname{Red}(Q)=n-1$.
2. $\operatorname{Red}(Q)=n-1$, and every vertex of $Q$ is the last remaining green vertex after some general maximal green sequence.
3. $\operatorname{uRed}(Q)=n-1$.
4. $\operatorname{uRed}(Q)=n-1$, and every vertex of $Q$ is the last remaining green vertex after some general reddening sequence.

The paper proves part (3), while the stronger variations in parts (1), (2), and (4) remain unresolved. Affirming any part would imply the mutation-invariance of admitting a reddening sequence.