Failure of the index theorem in an incomplete market economy

Recently, it was proved that the index of an economy with incomplete real asset markets is typically +1$+1$ when the degree of incompleteness, which is defined as the difference between the number of states and the number of securities, is an even number. This paper considers the case where the degree of incompleteness is an odd number and proves that any odd number can be realized as the index of such an economy.