**Question**: What is the largest Beraha number known to be an accumulation point of real zeros of the chromatic polynomial of planar graphs?

**Background**:
The Beraha numbers $B_n=2+2cos(2\pi/n), n=2,3,\ldots$ are well known in the study of the chromatic polynomial of planar triangulations, and more generally of planar graphs. Early results in the subject are due to W.T. Tutte: "On chromatic polynomials and the golden ratio" (J. Combinatorial Theory 9 (1970) 289–296) gave a bound on the value of the chromatic polynomial at $B_5={\phi}+1$, where $\phi$ is the golden ratio; "More about chromatic polynomials and the golden ratio" (1970 Combinatorial Structures and their Applications, 439–453) established the famous Tutte's golden identity.

My question concerns the observation of S. Beraha ("Infinite non-trivial families of maps and chromials", Thesis, Johns Hopkins University, 1975) that zeros of the chromatic polynomial of large planar graphs seem to accumulate to the numbers $B_n$. (The special role played by the Beraha numbers was also illustrated in the paper http://arxiv.org/abs/0711.0016 which gave a local relation satisfied by the chromatic polynomial of planar graphs at each $B_n$.)

There are different ways to interpret Beraha's conjecture; my question concerns accumulation points of *real* zeros.
In "Limits of chromatic zeros of some families of maps" (J. Combin. Theory Ser. B 28 (1980), 52–65) S. Beraha, J. Kahane and N.J. Weiss gave examples of (periodic) graphs where $B_5$ and $B_7$ are accumulation points.

I am wondering whether there are higher Beraha numbers that are also known to be accumulation points of real zeros. A more precise question concerns periodic graphs, like the ones considered by Beraha-Kahane-Weiss. Is there a conjecture one way or another?

[It is known (G.F. Royle, "Planar triangulations with real chromatic roots arbitrarily close to four", Ann. Comb. 12 (2008), 195–210) that there are graphs for which real zeros accumulate to 4, although I believe these examples are not "periodic". Also, the chromatic polynomial has been studied for various families of planar graphs in the context of statistical mechanics; I do not know whether this exhibited $B_n$, $n>7$ as accumulation points.]