### Counterexample
Suppose that in the honest case, the winner is $i$ and the loser is $j$, and $(s_i, c_i)$ and $(s_j, c_j)$ are their true optimal solutions, and these are the only two solvers in the competition. Let $c_i = 5$, $c_j = 10$, $Q(s_i) = 100$, $Q(s_j) = 100$. Then we have that the profit of player $i$, if they report honestly, is
\begin{align*}
p_i - c_i & = c_i + \left( \mathcal U(s_i, c_i) - \mathcal U(s_j, c_j) \right) - c_i \\
& = \mathcal U(s_i, c_i) - \mathcal U(s_j, c_j) \\
& = (Q(s_i) - c_i) - (Q(s_j) - c_j) \\
& = (100 - 5) - (100 - 10) \\
& = 5.
\end{align*}

Now suppose that player $i$ instead reports dishonestly, with $c_i' = 0$. Their profit becomes
\begin{align*}
p_i - c_{i'} & = c_{i'} + \left( \mathcal U(s_i, c_{i'}) - \mathcal U(s_j, c_j) \right) - c_{i'} \\
& = \mathcal U(s_i, c_{i'}) - \mathcal U(s_j, c_j) \\
& = (Q(s_i) - c_{i'}) - (Q(s_j) - c_j) \\
& = (100 - 0) - (100 - 10) \\
& = 10.
\end{align*}

We have found a way for player $i$ to misreport their cost and receive more profit as a result. This is a grave problem, rendering our original mechanism entirely invalid, because it leads to an equilibrium of all solvers reporting cost equal to 0, since they will either (a) win the competition and maximize their profits or (b) lose the competition and it won't matter that they lied.