This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 64 solutions found in 9.197 seconds (with a Bezout bound of 1458).
Reference: "Solving polynomial systems using continuation for engineering and scientific problems" by Alexander Morgan (p 148).
See also: http://homepages.math.uic.edu/~jan/Demo/eco8.html.
i1 : eco8(QQ)
o1 = {x x x + x x x + x x x + x x x + x x x + x x x + x x - 1, x x x
1 2 8 2 3 8 3 4 8 4 5 8 5 6 8 6 7 8 1 8 1 3 8
------------------------------------------------------------------------
+ x x x + x x x + x x x + x x x + x x - 2, x x x + x x x + x x x
2 4 8 3 5 8 4 6 8 5 7 8 2 8 1 4 8 2 5 8 3 6 8
------------------------------------------------------------------------
+ x x x + x x - 3, x x x + x x x + x x x + x x - 4, x x x +
4 7 8 3 8 1 5 8 2 6 8 3 7 8 4 8 1 6 8
------------------------------------------------------------------------
x x x + x x - 5, x x x + x x - 6, x x - 7, x + x + x + x + x +
2 7 8 5 8 1 7 8 6 8 7 8 1 2 3 4 5
------------------------------------------------------------------------
x + x + 1}
6 7
o1 : List
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The object eco8 is a method function.