Calculation of an amplitude using the Grassmannian approach (amplituhedron)

In the following review paper on scattering amplitudes, by Elvang and Huang:
https://arxiv.org/abs/1308.1697

they calculate the amplitude for 6 particles with half of positive helicity and half of negative helicity in section 9.3.2. Their matrix C (a point in the relevant Grassmannian) is parametrized as in Eq. 9.35.
They then use a contour of integration giving three contributions (after applying the residue theorem). The three contributions are given in 9.45 and 9.46.

Now, Arkani-Hamed et al do the same calculation using a slightly different approach, see https://arxiv.org/pdf/1212.5605.pdf. They use plabic graphs to represent different contributions to an amplitude. The plabic graphs corresponding to the amplitude of interest are given in Figure 16.6, with the total amplitude given just below, in Eq. 16.7.

Continuing with Arkani-Hamed et al, the contribution from one of these three plabic graphs is calculated (without providing much details) in Eqs 8.6 and 8.7. They give now the matrix C once the integral is done. Now, they don’t show it but they do mention (just below 8.7) that they use a parametrization "making the residue about the pole (123)=0 easy to read off" (here, (123) is the minor involving the first three columns of C). So the matrix C they give in (8.6) is the result obtained after doing the integral.

Now I can ask my question: How are the two approaches related? In the case of Elvang and Huang, they have a single integral which generates all three terms in the full amplitude. In the case of Arkani-Hamed et al, they have three plabic graphs that each apparently contribute a single term in the amplitude.

So the initial integral used by Elvang and Huang must differ from the integral used by Arkani-Hamed et al. It is difficult to see because of the different notation used and the different ways of writing the delta functions. I am hoping that someone who knows this material will be able to point out how the two approaches differ, at a basic level. A related question is: then it does not seem that one can associate to the intergal a plabic graph to the approach followed by Elvang and Huang since there single integral gives all three plabic graphs. Is that right?

Thanks in advance