The cubic plus-equation solution conjecture

Let $a,b,x$ be positive integers with $a<b$, and consider

$$(a^3+1)(b^3+1)=x^2.$$

Cubic plus-equation conjecture. The only positive integer solutions are

$$(a,b,x)\in\{(1,23,156),(6,26,1953),(20,362,616077)\}.$$

This conjecture is motivated by a computer search with $n=3$ and $1\leq a<b\leq10^6$. The source does not state that the search establishes completeness beyond that range, and its general resolution status is not given.