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 136 solutions found in 4.954 seconds (with a Bezout bound of 216).
Reference: PoSSo test suite.
See also: http://homepages.math.uic.edu/~jan/Demo/rose.html.
i1 : rose(QQ)
4 20 2 2 4 7 4 7 4 50 2 35 49 3 6 2 5 3
o1 = {y - --x , x z + --x*z + --z - --x - --x - ---, -x y z + x y +
7 10 48 27 27 216 5
------------------------------------------------------------------------
3 5 2 7 4 3 7 4 2 3 4 3 609 3 3 63 3 2 77 3 2
-x y z + -x y - --x y*z - --x z + ----x y + ---x y z - ---x y*z -
7 5 20 20 1000 200 125
------------------------------------------------------------------------
21 3 3 49 2 3 147 2 2 23863 2 2 91 2 3 27391 3
--x z + ----x y + ----x y z - -----x y*z - ---x z - ------x*y +
50 1250 2000 60000 400 800000
------------------------------------------------------------------------
4137 2 1078 2 5887 3 1029 3 24353 2
------x*y z - ----x*y*z - ------x*z - ------y - -------y*z -
800000 9375 200000 160000 1920000
------------------------------------------------------------------------
343 3
------z }
128000
o1 : List
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The object rose is a method function.