On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$

We study the Diophantine equation $(a^n+1)(b^n+1)=x^2$, which belongs to the family of equations originating from the work of Szalay in 2000.

If $a>1$, it is shown that the equation of the title has only one solution in positive integers, when $a$ and $b$ are distinct powers of the same integer $t>1$.

Also, a complete description of the solutions is obtained under the assumptions that $a$ and $b$ are coprime and $n$ is even.

Several other special cases of the equation are considered, and two conjectures are proposed.