The monotonicity conjecture for quotients of squared Bessel zeros

Let $j_{1,k}$ denote the $k$th positive zero of the Bessel function of the first kind of order $1$, with the convention $j_{1,0}=0$, and define

$$\hat q_k=\frac{j_{1,k+1}^2-j_{1,k}^2}{j_{1,k}^2-j_{1,k-1}^2}.$$

The Bessel-zero quotient monotonicity conjecture. The quotients $\hat q_k$ decrease monotonically as $k\in\mathbb{N}$ increases. The conjecture is supported by numerical experiments, but no resolution is given in the source.