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Let $k$ be a positive integer, and let $a,b$ be coprime positive integerswith $\min\{a,b\}>1$. In this paper, using a combination of some elementarynumber theory techniques with classical results on the Nagell-Ljunggrenequation, the Catalan equation and some new properties of the Lucas sequence(\seqnum{A000204} in OEIS), we prove that if $k>1$ and $a,b$ are both primepowers with $\min\{a,b\}>2$, then the equation $(ak)^x+(bk)^y=((a+b)k)^z$ hasonly one positive integer solution $(x,y,z)=(1,1,1)$. The above resultpartially proves that Conjecture 1 presented in (Acta Arith. 2018, 184 (1):37-49) is true.

On the Ternary Purely Exponential Diophantine Equation $(ak)^x+(bk)^y=((a+b)k)^z$ with Prime Powers $a$ and $b$