The lower bound (initially provided by Khandawit and Sriwasdi) was improved in 2009 by Dimitrios Pagonakis. Although the paper has not been put on the arxiv yet (I understand that the author has plans to do this), the bound was improved from 0.227498 to 0.231999.
Cf. Page 8/12 in http://www-math.mit.edu/news/summer/2009RSIAbstracts.pdf