Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?

The height of a rational number p/q is denoted by h(p/q) and equals max (|p|, |q|) provided is written in lowest terms. The height of a rational tuple h(x₁,..., x_n) is denoted by h(x₁,..., x_n) and equals max(h(x₁),..., h(x_n)). Let G_n = {x_i + 1 = x_k : i, k ϵ {1,..., n}} ∪ (x_i, · x_j = x_k : i, j, k ϵ {1,..., n}}. Let f (1) = 1, and let f (n + 1) = 2^9f(n) for every positive integer n. We conjecture: (1) if a system S ⊆ G_n has only finitely many solutions in rationals x₁,..., x_n, then each such solution (x₁,..., x_n) satisfies h(x₁,..., x_n) ≤ {1 (if n = 1) 2^2n-2 (if ≥ 2) if a system S ⊆ G_n has only finitely many solutions in non-negative rationals x₁,..., x_n, then each such solution (x₁,..., x_n) satisfies h(x₁,..., x_n) ≤ f (2n). We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution.