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 146 solutions found in 6.540 seconds (with a Bezout bound of 720).
Reference: PoSSo test suite.
See also: http://homepages.math.uic.edu/~jan/Demo/reimer5.html.
i1 : reimer5(QQ)
2 2 2 2 2 3 3 3 3 3 4
o1 = {2x - 2y + 2z - 2t + 2u - 1, 2x - 2y + 2z - 2t + 2u - 1, 2x -
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4 4 4 4 5 5 5 5 5 6 6
2y + 2z - 2t + 2u - 1, 2x - 2y + 2z - 2t + 2u - 1, 2x - 2y +
------------------------------------------------------------------------
6 6 6
2z - 2t + 2u - 1}
o1 : List
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The object reimer5 is a method function.