Irreducible density race : ring $A = \Bbb Z[\frac{1+\sqrt {-23}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-31}}{2}]$?

Question

Inspired by this question here :

Prime density race : ring $A = \Bbb Z[\frac{1+\sqrt {-7}}{2}]$ vs ring $B = \Bbb Z[\frac{1+\sqrt {-11}}{2}]$?

I now consider the smallest cases of non-UFD. I did not skip class number 2, it appears to be a fact that they have class number 3.

Anyways,

Consider the ring $A = \Bbb Z[\frac{1+\sqrt {-23}}{2}]$ with the elements $\frac{a+b\sqrt {-23}}{2}$ where $a,b$ are both even or both odd integers.

This is also known as the ring of integers in the imaginary quadratic field $\Bbb Q[\sqrt {-23}]$.

Consider also the ring $B = \Bbb Z[\frac{1+\sqrt {-31}}{2}]$ with the elements $\frac{c+d\sqrt {-31}}{2}$ where $c,d$ are both even or both odd integers.

This is also known as the ring of integers in the imaginary quadratic field $\Bbb Q[\sqrt {-31}]$.

Both $A$ and $B$ are non-UFD with class number $3$.

Consider $f_{23}(r)$ as the number of irreducibles in $A$,such that $1 < a^2 + b^2 < r$ and likewise

Consider $f_{31}(r)$ as the number of irreducibles in $B$,such that $1 < c^2 + d^2 < r$.

How to show that

$$ g(r) = f_{23}(r) - f_{31}(r)$$

1. changes sign infinitely often

2. is unbounded for both positive and negative output

3. is equal to zero infinitely often

Plots are welcome too !