Terai's conjecture with prescribed exponents

Let $b,c,r,t$ be fixed positive integers satisfying

$$b^r+1=2c^t,\qquad \min\{b,c\}>1,\qquad t\in\{1,2\}.$$

Let $x,y,z\in\mathbb{N}$ satisfy $x^2+b^y=c^z$. Terai's prescribed-exponent conjecture. The only solution is

$$(x,y,z)=(c^t-1,r,2t).$$

The conjecture has been verified in several special cases, but the source does not state a general resolution.