I am the aforementioned economist. I figured out a counterexample for the non-singular case:

Let $$ A = \left( \begin{array}{ccc} 0 & 3 & 1 \\ 3 & 0 & 1 \\ 2 & 2 & 2 \end{array} \right) $$

$$ z = \left( \begin{array}{ccc} \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} \end{array} \right) $$

$$ x = \left( \begin{array}{ccc} 0, & \frac{1}{2}, & \frac{1}{2} \end{array} \right) $$

$$ y = \left( \begin{array}{ccc} \frac{1}{2}, & 0, & \frac{1}{2} \end{array} \right) $$

The expected payoff is 2 in the symmetric equilibrium and $\frac{3}{2}$ in the other one. As far as I can tell there is no 2x2 non-singular counterexample.

I am the aforementioned economist. I figured out a counterexample for the non-singular case:

Let $$ A = \left( \begin{array}{ccc} 0 & 7 & 0 \\ 7 & 0 & 0 \\ 3 & 3 & 1 \end{array} \right) $$

$$ z = \left( \begin{array}{ccc} \frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} \end{array} \right) $$

$$ x = \left( \begin{array}{ccc} \frac 15, & 0, & \frac45 \end{array} \right) $$

$$ y = \left( \begin{array}{ccc} 0, & \frac 15, & \frac45 \end{array} \right) $$

The expected payoff is $2\cdot \frac 73$ in the symmetric equilibrium and $\frac 75 + \frac 75$ in the other one. As far as I can tell there is no 2x2 non-singular counterexample.