Modified prime number theorem for the number trail

Let $p_k$ be the $k$th prime and define the number-trail prime-counting function by

$$\pi_\infty(N):=\max\{k:L_\infty(p_k)\leq N\}.$$

Let $c_0$ be the constant defined by the limiting relation for $L_\infty(N)/N$, and let

$$\operatorname{Li}(x)=\int_2^x\frac{1}{\ln y}\,dy.$$

Modified prime number theorem.

$$\lim_{N\to\infty}\frac{\pi_\infty(N)}{N/\log N} =\lim_{N\to\infty}\frac{\pi_\infty(N)}{\operatorname{Li}(N)} =\frac{1}{c_0}\approx0.436992\ldots.$$

This conjecture predicts that counting primes along the number trail changes the usual prime-number asymptotic only by the scaling factor $1/c_0$; it is conditional in the surrounding discussion on the existence of the limiting constant and is not resolved in the source.