The two-prime -primitive number conjecture

Let $p$ and $q$ be distinct primes, and let $x,y\in\mathbb{N}$. Consider natural numbers of the form

$$n=p^xq^y.$$

A natural number is $\Delta$-primitive if it has the $\Delta$ property and is not obtained by multiplying a smaller $\Delta$-primitive number by a nontrivial square. Two-prime $\Delta$-primitive number conjecture. If $n$ is $\Delta$-primitive, then

$$n\in\{24,40\}.$$

The claim is motivated by experimental evidence and by the preceding theorem, which establishes the corresponding classification for numbers of the form $p^kq$; the general two-prime case remains open.