Hilbert's tenth problem for rings of holomorphic functions of bounded order

One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable.

In the direction of an answer, for every $\rho\ge 0$, we give a negative solution to the analogue of HTP in the ring of complex entire functions in one variable of growth order at most $\rho$.