A finite trellis with tied maxima and an unreachable high-score state

Take $T=3$, vertex layers $V_1=\{s\}$, $V_2=\{a,b,c\}$, and $V_3=\{t\}$. Set $\alpha (s)=1$ and $\beta (t)=1$. The only layer-2 edges are $s\to a$ and $s\to b$, each of weight $2$. The layer-3 edges are $a\to t$ and $b\to t$, each of weight $3$, and $c\to t$ of weight $100$. The vertex order in layer 2 is $a<b<c$.

Columns in the diagram are layers and arrow labels are multiplicative edge weights. The dashed vertex $c$ has no incoming edge, so it is unreachable. The proof below uses the explicit edge list and scores, not the drawing.

\begin{tikzpicture}[>=stealth]
\node[circle,draw] (s) at (0,0) {$s$};
\node[circle,draw] (a) at (2,1) {$a$};
\node[circle,draw] (b) at (2,0) {$b$};
\node[circle,draw,dashed] (c) at (2,-1) {$c$};
\node[circle,draw] (t) at (4,0) {$t$};
\draw[->] (s) -- node[above] {$2$} (a);
\draw[->] (s) -- node[above] {$2$} (b);
\draw[->] (a) -- node[above] {$3$} (t);
\draw[->] (b) -- node[below] {$3$} (t);
\draw[->] (c) -- node[below] {$100$} (t);
\end{tikzpicture}