The limiting constant for the $ell_\infty$ number-trail length

Let $L_\infty(N)$ denote the total length of the number trail through the prime grid from $1$ to $N$, measured in the $ell_\infty$ metric. The limiting-constant conjecture. The limit

$$\lim_{N\to\infty}\frac{L_\infty(N)}{N}=c_0$$

exists, with $c_0\approx 2.28836951\ldots$. This asymptotic claim is supported by computations, but no proof or resolution is given in the source.