A bivariate normal example where the locally most powerful and distantly most powerful test statistics are beaten everywhere from a median p-value standpoint

In the bivariate normal, n=2 case, when testing H₀:μ_x=μ_y=0,σ² _x=σ² _y=1, ρ=0 vs. H₁:μ_x=μ_y=0,σ² _x=σ² _y=1, 0<ρ<1, it is shown that the median p-values given by the locally most powerful test and the distantly most powerful test are both beaten everywhere by the median of a third test.