On a convexity problem on subfactors considered by Bhat, Pati and Sunder (Q1359521)

It has been shown in [\textit{R. Bhat}, \textit{V. Pati} and \textit{V. S. Sunder}, this Ann. 296, No. 4, 637-648 (1993; Zbl 0791.46006)] that for an inclusion \(N\subset M\) of type \(\text{II}_1\) factors with index \([M:N]=2\), if \(\partial_e(S)\) denotes the set of the extremal points of the set \(S=\{x\in M^+: E_N(x)= \lambda I\}\), where \(\lambda=[M:N]^{-1}\), and \(J(M,N)= \{e\in M: e\in{\mathcal P}(M), E_N(e)=\lambda I\}\) denotes the set of the Jones' projections, then one has the equality \(\partial_e(S)= J(M,N)\), and it has been conjectured that for a general irreducible inclusion, namely \(N'\cap M=\mathbb{C} I\), the same equality may hold. Using results of \textit{S. Popa} [Pac. J. Math. 137, No. 1, 181-207 (1989; Zbl 0699.46042)] on the relative dimension for projections, we disprove this conjecture by showing that for \([M:N]\neq 2\) the equality \(\partial_e(S)= J(M,N)\) never holds.