Congruence characterization of Lehmer-Euler numbers modulo powers of three

Prove that, for non-negative integers m, n and positive integers k, if 3^n is congruent to 3^m modulo 2·3^k, then the Lehmer-Euler numbers W_{3^n} and W_{3^m} are congruent modulo 3^{k+1}.

Background

Section 3.1 investigates congruence patterns for the Lehmer-Euler numbers W_{3n} modulo powers of three, motivated by an analogous theorem for Euler numbers modulo powers of two. The authors first note that the direct analogue, W_{3^n} ≡ W_{3^m} (mod 3^k) if and only if 3^n ≡ 3^m (mod 3^k), is false. They then examine computational periodicity and congruence patterns modulo 3^2, 3^3, 3^4, and 3^5.

Based on these results, the paper proposes a stronger conditional congruence: congruence of the powers 3^n and 3^m modulo 2·3^k should imply congruence of the corresponding Lehmer-Euler numbers modulo the next power, 3^{k+1}. The statement is presented as Conjecture 1 and is not proved in the paper.