Ma's conjecture on squares in exponential expressions

Let $p$ be an odd prime and let $b,t,r\in\mathbb{N}$. Define

$$Y=2^{2b}p^{2t}-2^{2b}p^{t+r}+1,$$

and

$$Z=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1.$$

Ma's conjecture. The following assertions hold:

(A) $Y$ is a square if and only if $t=r$, equivalently, if and only if $Y=1$.

(B) $Z$ is a square if and only if $p=5$, $b=3$, $t=1$, and $r=2$, equivalently, if and only if $Z=2401$.

This conjecture, presented by S. L. Ma in 1992, concerns the occurrence of squares in exponential expressions involving an odd prime. The source introduces it as an application of the Størmer theorem; no resolution status is supplied here.