Converting system of m rational ODEs into one n-order rational ODE preserving regularity

Question

Who knows anything about the Conjecture, which claims that for any explicit system of m 1st order rational ODEs

x'(t) = Rational RHS

y'(t) = Rational RHS

z'(t) = Rational RHS

. . . . . .

regular at a given point t, for any of its components, say x(t), there exists some n-order explicit rational ODE

x⁽ⁿ⁾ = Rational RHS

also regular at the same initial point t.

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Please see the links for the details:

http://TaylorCenter.org/gofen/UnifyingViewWithGap.pdf

http://TaylorCenter.org/gofen/UnifyingViewWithGap.pps

The Conjecture – in slides 6-8, a description of the gap in the theory – in slides 4, 3.

This Conjecture is obviously true if the class "rational" is changed to "holomorphic".

Also the Conjecture is proven true if we do not ask that the target n-order rational ODE be regular at the given point.