Does there Exist an Algorithm which to Each Diophantine Equation Assigns an Integer which is Greater than the Modulus of Integer Solutions,if these Solutions form a Finite Set?

Let E_n = {x_i = 1; x_i + x_j = x_k; x_i · x_j = x_k : i; j; k ∈ {1,...,n}}. We conjecture that if a system $S \subseteq E_n$ has only finitely many solutions in integers x₁,...,x_n, then each such solution (x₁,...,x_n) satisfies |x₁|,...,|x_n| ≤ 2²ⁿ⁻¹. Assuming the conjecture, we prove: (1) there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set, (2) if a set $\cal{M} \subseteq \mathbb{N}$ is recursively enumerable but not recursive, then a finite-fold Diophantine representation of $\cal{M}$ does not exist.