I will prove an upper bound $\mu \leq \tfrac{1}{\log 4}$. Fix a positive integer $r$. For $0 \leq k \leq r-1$, let $N_k$ be $2^{k/r} N$ rounded to the nearest integer, where $N$ is a large parameter. Considering the $N_{r-1}$ intervals present at time $N_{r-1}$, let $X_k$ be the number of them created between times $X_{k-1}$ and $X_k$; with $X_0$ the number created before time $0$. Then $$X_0 \frac{\mu}{N_0} + X_1 \frac{\mu}{N_1} + \cdots + X_{r-1} \frac{\mu}{N_{r-1}} \leq 1.\quad (\ast)$$
Of the $N_k$ intervals present at time $N_k$, at most $N_{r-1} - N_k$ can be destroyed by time $N_{r-1}$. So at least $2 N_k-N_{r-1}$ of them survive until time $N_{r-1}$. This means that $$X_0+X_1+\cdots + X_k \geq 2 N_k - N_{r-1} \quad (\dagger),$$ with equality for $k=r-1$.
The coefficient of $\mu$ on the left hand side of $(\ast)$ is linear in the $X_i$, so it is minimized if we take the $X_i$ at a vertex of the polytope cut out by $(\dagger)$ and $X_i \geq 0$. A little thought shows we want the vertex where $(\dagger)$ is equality for all $k$, so $X_0 = 2 N_0 - N_{r-1}$ and $X_k = 2 N_k - 2 N_{k-1}$ for $1 \leq k \leq r-1$. We deduce $$\mu \left( \frac{2N_0-N_{r-1}}{N_0} + \frac{2 N_1 - 2 N_0}{N_1} + \cdots + \frac{2 N_{r-1} - 2 N_{r-2}}{N_{r-1}} \right) \leq 1.$$
We plug in our optimized values: $$\mu \left( 2r - r 2^{(r-1)/r} \right) \leq 1$$ so $$\mu \leq \frac{1/r}{2 (1-2^{-1/r})}.$$
Finally, taking the limit as $r \to \infty$ gives $\mu \leq \tfrac{1}{\log 4}$ as promised.