Given a mixed state with a $n$ qubit density matrix of the following structure: $$ \rho=\pmatrix{\lambda_1&&&&\nu\\&\lambda_2\\ &&\lambda_{\dots}\\ &&&\lambda_2\\ \nu&&&&\lambda_1}, $$ so a mixture of a GHZ state and a sum of diagonal contributions of the form: $\prod Z_k^{x_k}$, where $x_k$ is the $k$th bit of the binary number $x\in\{0,1\}^n$, which is restricted to contain an even number of $1$s. The latter restriction ensures that the matrix is symmetric w.r.t. the anti-diagonal.
If all $\lambda_k$ are equal, there is a way to measure exponentially small amounts of the GHZ state, as proposed here. It looks to me, that my states don't respect the graph diagonal state requirement given in the comments.
What is the best lower bound for $\nu$, that is efficiently measurable (involving polynomially many measurements)?