The asymptotic floor-sum estimate for primes $p=4n-1$

Let $n$ range over positive integers such that $p=4n-1$ is prime, and let $M$, $R_m$, and $Q_m$ be the quantities defined earlier in the paper. Asymptotic floor-sum conjecture.

$$\lim_{n\rightarrow\infty}\frac{\displaystyle\sum_{m=1}^{M}\left\lfloor R_m\right\rfloor+\displaystyle\sum_{m=0}^{M-1}\left\lfloor Q_m\right\rfloor}{Mp+2n}=\frac{1}{3}.$$

Moreover, a good estimate for the numerator is

$$\left\lfloor\frac{Mp+2n}{3}\right\rfloor.$$

The claim is motivated by numerical data accompanying the paper's class-number identities. No resolution is supplied in the given text.