Lakshminarayan's real-eigenvalue conjecture for products of random matrices

Let $X_1,X_2,\dots,X_n$ be i.i.d. matrices of size $k\times k$, whose entries are i.i.d. real random variables distributed according to a probability measure $\mu$, and let

$$A_n=X_1X_2\dots X_n.$$

Lakshminarayan's conjecture. The probability that $A_n$ has all real eigenvalues tends to $1$:

$$\lim\limits_{n\rightarrow\infty}\mathbb{P}(A_n\text{ has all real eigenvalues})=1.$$

For i.i.d. real Gaussian entries, this conjecture was proved by Forrester and independently by Nanda, with exponential convergence to $1$. The stated version asks for the same phenomenon for arbitrary entry distributions $\mu$; the paper proves it when $\mu$ has an atom.