The statement referenced by Igor Rivin http://www.math.clemson.edu/~janoski/ResearchStatement.pdf uses the phrase
Computationally looking at p(n) we see that for n ≥ 26 the partition function is log-concave [2].
I had seen this reference before probably about the same time this research statement was first released, and I am skeptical for two reasons.
The phrasing "Computationally..." would seem to indicate some type of calculation. This cannot involve a computer since it would have to hold for all n larger than 26, and I am not aware of any simplification that allows one to only consider a finite number of cases. It would have been helpful to at least expound on the type of computations involved.
I checked for the promised reference, and indeed I found it on the CV of the author, http://www.math.clemson.edu/~janoski/VitaTex.pdf, but it refers to the quote below. I did a quick google search and I could find no reference or anything pointing to a publication.
Brian Bowers, Neil Calkin, Kerry Gannon, Janine E. Janoski, Katie Joes, Anna Kirkpatrick, The Log Concavity of the Partition Function, (in preparation)
Asymptotics will not provide the answer here, since n sufficiently large doesn't hold up unless you can provide a concrete n and test everything less than it, and I don't believe the Hardy-Ramanujan asymptotic expansion yields any guaranteed error estimates.
It may be possible to use DH Lehmer's estimates to obtain a proof. In two papers (1937 and 1939) he investigated the coefficients of both the Hardy-Ramanujan asymptotic expansion and the Hardy-Ramanujan-Rademacher expansion. He provided guaranteed error bounds on the remainder terms in the asymptotic expansions so that, for example, his Theorem 13 says that for n>600, only $2/3 \sqrt n$ terms of the Hardy-Ramanujan asymptotic series are needed to estimate p(n) to the nearest integer.
At present, I don't believe the matter is completely settled, despite the overwhelming computational evidence.