The frequency conjecture for ratios arising from $p_{m-1}+1$

Let $p_m$ denote the $m$-th prime, set $n_m=p_{m-1}+1$, and let $L_m$ be the largest prime factor of $n_m$ when $L_m>m$; define $R_m=n_m/L_m$. For a fixed positive integer $r$, write $\varphi(r)$ for Euler's totient function. Frequency conjecture. For any fixed even integer $r\geq 2$, there is a positive constant $C$ close to $1$ such that

$$\#\{m\leq N:R_m=r\}\sim \frac{CN}{\varphi(r)\log N}.$$

For odd values of $r$, the count is identically zero for all $m\geq 3$. The heuristic comes from requiring $r\mid p_{m-1}+1$ and the remaining quotient to be prime; the evenness restriction follows from the parity of $p_{m-1}+1$. Establishing the asymptotic, including the nature of the constant $C$, remains open.