Bank's hypertranscendency conjecture for perturbations of the gamma function

Let $h$ be an entire function, let $\Gamma$ denote the Euler gamma function, and let $\mathcal{M}_0$ be the field of meromorphic functions $y$ satisfying $T(r,y)=o(r)$ as $r\rightarrow\infty$ outside a set of finite measure. Bank's conjecture. The function

$$\Gamma e^h$$

should be hypertranscendental over $\mathcal{M}_0$, meaning that it satisfies no nontrivial algebraic differential equation with coefficients in $\mathcal{M}_0$. Bank posed this question to determine whether the hypertranscendency of $\Gamma$ is caused by the nature of its poles and zeros; the conjecture is stated here without evidence of a resolution.