Asymptotic distribution conjecture for the floor-function sum

Let $f(n)$ denote the sum defined earlier in the paper. The limiting-values conjecture. As $n\to\infty$,

$$\lim_{n\to\infty}\frac{f(4n)}{4n}=\frac18,$$ $$\lim_{n\to\infty}\frac{f(4n+1)}{4n+1}=0,$$ $$\lim_{n\to\infty}\frac{f(4n+2)}{4n+2}=-\frac18,$$ $$\lim_{n\to\infty}\frac{f(4n+2)}{4n+3}=-\frac14.$$

The paper motivates these limits through numerical plots and says they are supported by a mixture of proven and conjectured identities, but does not prove the limits themselves.