 Simplex paths on a cube with one truncated corner

For $0\le x_i\le 1$, $x_1+x_2+x_3\le 5/2$, and objective $x_1+2x_2+4x_3$, the oriented edge graph has eight improving paths from the origin to the unique optimum $(1/2,1,1)$. Their lengths range from three to five pivots.