Monotonic-zero conjecture for incomplete gamma functions

Let $a>0$ be a fixed real number, and let $Q$ denote the first quadrant of the complex plane. A function has monotonic zeros in $Q$ when its zeros there, listed by increasing real part, have monotone nondecreasing imaginary parts. Incomplete-gamma monotonic-zero conjecture. The incomplete gamma function $\Gamma(z,a)$ has monotonic zeros in $Q$. The source says that computer calculations support this conjecture, while the general zero distribution of $\Gamma(z,a)$ for fixed positive real $a$ is not well understood.