(This is a follow-up to The spectral radius of a binary matrix)

Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.

**QUESTION.** Is there a $\delta\in\bigl(0,\frac12\bigr), \alpha\in(0,1)$ and $b>0$ such that for any $n\ge1$ and any $M\in\mathcal B_n$ with the number of 0s less than $\delta n^2$ we have $\rho(M)\ge bn^\alpha$?

If so, what's the best known value of $\delta$?

The **motivation** for this question comes from symbolic dynamics (hence the tags). Namely, is it true that for any topological Markov chain given by a matrix with "sufficiently many" 1s its topological entropy is positive?