I am defining a "somewhat normal number" as such:

A somewhat normal number is a number where every possible numerical string in base $10$ can be found in said number at least once, but the density of each $n$ digit string may not be equal to $10^{-n}$.

A somewhat normal number is a weaker version of a normal number, since it removes the requirement of the density of each $n$ digit string being equal to $10^{-n}$.

An example of a number that is somewhat normal but not normal would be $$0.\color{red}11\color{red}21\color{red}31\color{red}41\color{red}51\color{red}61\color{red}71\color{red}81\color{red}91\color{red}{10}11\color{red}{11}11\color{red}{12}11\cdots$$

i.e. every natural number (highlighted in red) with $1$s added to the end of each number is concatenated together (like in the Champernowne constant), and the number of $1$s is equal to the length of the natural number preceding it.

The number contains all of the natural numbers and therefore each possible numberical string can be found in this number. However there are $1$s than other digits in this number, and this number is therefore not normal.

Under this definition of "somewhat normal", can we say that $\pi$ is a somewhat normal number?