On a conjecture concerning the exponential Diophantine equation $(a{n}^2+1{)}^x+(b{n}^2-1{)}^y=(cn{)}^z$

Let $a$, $b$, $c$, and $n$ be positive integers such that $a+b=c^2$, $2\nmid c$ and $n>1$. In this paper, we prove that if $gcd(c,n)=1$ and $n\ge 117.14c$, then the equation $(a{n}^2+1{)}^x+(b{n}^2-1{)}^y=(cn{)}^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$ under the assumption $gcd(a{n}^2+1,b{n}^2-1)=1$. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer $n$, if $(a,b,c)=(12,13,5)$, $(18,7,5)$, or $(44,5,7)$, then this equation has only the solution $(x,y,z)=(1,1,2)$. This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.