The Paradox (Hypothesis) of Van Petrov's 'Impossibility' Equation

[syndixxx]

There is a statement recently published in the author's note:

For this complex equation with natural variables (distinct and greater than one), there is no general algorithm to prove or disprove the existence of a solution, except by complete enumeration. Moreover, attempting to prove that no such algorithm exists, as well as attempting to prove this statement, will prove more difficult than simply performing the enumeration.

k ^ n * (a^n + b^m) / (n ^ a + c) = n^k * sin(m ^ n + b ^ a) / (m ^ a - c),
where a, b, c, n, m, k ∈ N, a, b, c, n, m, k > 1, and a ≠ b ≠ c ≠ n ≠ m ≠ k.

[If the translation is not accurate, I am attaching the original publication.]

I wonder how true this statement is and how one could try to prove or disprove it? As I understand it, this is essentially a type of Diophantine equation?