= Solution
An acyclic finite <quiver> has integer vertex weights $w_i$ such that $w_j>w_i$ for every arrow $i\to j$; for example, let $w_i$ be the maximum length of a path ending at $i$. For $t\ne0$, act by $g_i(t)=t^{w_i}I_{n_i}$. Then
$$
(g(t)\cdot f)_\rho=t^{w_{t(\rho)}-w_{s(\rho)}}f_\rho.
$$
Every exponent is positive. The arrow matrices therefore extend polynomially to $t=0$ with all arrows zero, which is the origin of the <quiver representation space>. The values for $t\ne0$ lie in $\mathcal O_X$, so
$$
\boxed{0\in\overline{\mathcal O_X}\quad\text{for every }X.}
$$
This proves that <acyclic quiver representations degenerate to zero>, also when some vertex spaces are zero.
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