Certain it is that $1$ will be the most common initial digit. Consider the relation
$f(n,d)=\int_{d×10^n}^{(d+1)×10^n}\dfrac{dt}{\ln t}$
in which the integrand reflects tge asymptotic density of primes. The asymptotic density is monotonically decreasing, so for any value of $n$ sufficiently large for the asymptotic relation to hold accurately the count of primes will be greater for $d=1$ than for any of $d=2,d=3,$ etc.