Minkowski dimension conjecture for degenerate-focus spirals

Let $k$ be a positive integer, and let $m>n$ with $m$ and $n$ odd. Consider a spiral trajectory $S$ of the degenerate-focus system near the origin $(x,y)=(0,0)$.

Minkowski dimension conjecture. Any such spiral trajectory has Minkowski dimension

$$\dim_B S=2-\frac{1+\frac{n}{m}}{1+2kn}.$$

This conjecture was proposed based on numerical observations for the case $m>n$ with $m,n$ odd. The surrounding results establish comparability with a power spiral but do not determine the precise Minkowski dimension in this case.