Laradji–Mignotte–Tzanakis conjecture for $5x^2+q^{2\alpha}=y^5$

Let $q$ be an odd prime, and consider integer solutions $(x,y,\alpha)$ with $x,y,\alpha>0$ of

$$5x^2+q^{2\alpha}=y^5.$$

Laradji–Mignotte–Tzanakis conjecture. If $q$ is not of the form

$$q=2000v^4-200v^2+1$$

for an integer $v$, then the equation has no solutions. This conjecture strengthens the cited theorem's conclusion in the remaining case after the theorem's necessary parametrization and is motivated by experimental computations; its resolution is not stated in the source.