Longest-run conjecture for Ramanujan and non-Ramanujan primes

Let $p_N$ be the $N$th prime. For each $N$, consider the longest consecutive run, among the primes at most $p_N$, consisting entirely of Ramanujan primes; define analogously the longest run of non-Ramanujan primes.

Longest-run conjecture.

$$\limsup_{N\to\infty}\frac{\text{length of the longest run of Ramanujan primes among primes}\le p_N}{\log N/\log 2}\ge1,$$

and the same inequality holds when “Ramanujan” is replaced by “non-Ramanujan”.

The prediction strengthens an earlier claim that arbitrarily long runs of both types exist. Numerical data in the paper support the stated logarithmic lower bound, but the supplied text gives no resolution.