TOKYO, June 1 (Reuters) - ValueAct Capital has called for private dialogue with Seven & i Holdings Co Ltd after the U.S. hedge fund lost a public battle to change the leadership of the Japanese retail giant. The fund, in a letter to shareholders dated Wednesday, said it looked forward to "private and constructive dialogue" with the operator of the 7-Eleven and Speedway convenience store chains. Seven & i shareholders last week voted down all four of ValueAct's board nominees, dismissing the fund's attempt to replace President Ryuichi Isaka and other company leaders. Still, ValueAct said the number of votes its nominees garnered showed clear desire for change, and that underperformance of Seven & i stock since the shareholder meeting indicated investor disappointment with the status quo. "Now, all shareholders - including ValueAct - are in a position to hold the entire Board of Directors accountable," the fund said in its letter.