Reference Request : The number of binary string with some special condition (open problem)

Question

(Sorry for my poor english...)

I wonder the reference of Simon Marais Mathematics competition 2019 problem B4. This problem is as follow.

(They said this problem is open problem.)

B4. A set $\mathcal{B}$ of binary strings is defined following rules.

1. 1 in $\mathcal{B}$.

2. If $s_1s_2\dots s_n$ in $\mathcal{B}$ and $n$ is odd, then $s_1s_2\dots s_n 0$ and $0s_1s_2\dots s_n$ are in $\mathcal{B}$.

3. If $s_1s_2\dots s_n$ in $\mathcal{B}$ and $n$ is even, then $s_1s_2\dots s_n 1$ and $1s_1s_2\dots s_n$ are in $\mathcal{B}$.

Let $b_n$ be the number of strings in $\mathcal{B}$ with length $n$.

Then, determine $\liminf_{n\to \infty} (b_n)^{1/n}$ and $\limsup_{n\to \infty} (b_n)^{1/n}$.

I want to find the reference of this problem...

Is there any reference related to this problem?

Here are the problem statement and the solution to part $(a)$ of the problem, which asks to show that there exist constants $c_1,c_2\gt0$ and $1.6\lt\lambda_1,\lambda_2\lt1.9$ such that $ c_1\lambda_1^n\lt b_n\lt c_2λ_2^n$ for all positive integers $n$.