I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant symplectic structures there are on the same space, i.e. $M := SU(3)/T^2$. Now as I understand it, there is a unique equivariant K"ahler metric for $M$, which will of course give us one equivariant symplectic structure, and since symplectic structures which do not arise from K"ahler metrics are hard to come by, I would conjecture that this is the only equivariant symplectic structure for this space. Is this true?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant symplectic structures there are on the same space, i.e. $M := SU(3)/T^2$. Now as I understand it, there is a unique equivariant K"ahler metric for $M$, which will of course give us one equivariant symplectic structure, and since symplectic structures which do not arise from K"ahler metrics are hard to come by, I would conjecture that this is the only equivariant symplectic structure for this space. Is this true?