The notorious Conjecture amounts to Inverse Function <<Theorem>>
of Algebric Geometry, which dates from 1939 but was tacitly assumed (as
an obvious Statement) by well-known mathematicians as late as the '60s.
After reviewing some well-known facts (mostly 2 decades old), I shall
dwell on the new approach (as in the title & what follows):

        The following `vague' question is prompted from what we know
already for  m=2:
        `HOW is it possible for a set of  n  polynomials in  m  variables,
all homogeneous of same degree  (d, say), to violate the inequality

         n < (m+k-1)-choose-k

and yet satisfy 3 requirements:  certain minimailties for  n  and  m, as
also their being solutions of a Linear DE (with polynomial coefficients)
which is homogeneous of order  k?'

        This is needed only for  m=n  case, with  d=3, to get ``JC''.