Consider a 2-player symmetric game given by a payoff matrix $A\in [0,1]^{n,n}$ for the row player (i.e. the column player matrix is $A^t$).

Let $s=<s_1,s_2>$ be a (possibly mixed-strategies, not-necessarily symmetric) equilibrium for the game.

Define $Sup(s)=Sup(s_1)\cup Sup(s_2)$, where $Sup(s_i)$ is the set of strategies in $s_i$ that has a strictly positive probability.

Is there always a symmetric equilibrium $<s',s'>$ such that $s'\subseteq Sup(s)$?

(note that there always exist a symmetric equilibrium for the game, but I'm interested in knowing whether any equilibrium imply a symmetric equilibrium with subset support).