Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question

Is every minimal hypersurface (i.e. codimensional one) in $S^n$ algebraic (i.e. given by intersecting the zero set in $R^{n+1}$ of some homogenous polynomial in n+1 variables with $S^n$) ?

e.g. the Lawson surface in $S^3$ is given by the imaginary part of $(x_1+ix_2)^m(x_3+ix_4)^l$.

What is the current status of this problem?