Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \ast A$).

Is it true that for any such sequence $A$, there exists $n$ such that $A^{\ast n}$ is log-concave?

As an example, any sequence of the form $A = (1,x,1)$ is log concave if $x\geq 1$, and $A\ast A$ is log concave if $x\geq \sqrt{2/3}$. In general, I have checked numerically that $A^{\ast k}$ is log concave if $x\geq \sqrt{2/(k+1)}$. I have also checked this statement numerically for many other sequences of longer length, but I have no idea how to go about proving anything.