The Rebate Edge

Shipping cost routing. Edge weights are net cost in dollars, and one carrier pays a rebate, so that leg is negative. Directed edges:

s→a 4, a→x 5, s→b 11, b→x -8, x→t 1

There is no cycle at all, so certainly no negative cycle. This is the textbook heap Dijkstra, which marks a vertex done when it is popped:

// q is a min-heap keyed on distance
while (!q.empty()) {
    auto [d_v, v] = q.top(); q.pop();
    if (done[v]) continue;
    done[v] = true;
    for (auto [to, w] : adj[v]) {
        if (dist[v] + w < dist[to]) {
            dist[to] = dist[v] + w;
            q.push({dist[to], to});
        }
    }
}

What does it leave in dist[t], and what is the true cheapest cost?