Unweighted Gaussian-to-spherical design transfer

Determine whether there exists, for each fixed t, a constant c_t such that every unweighted Gaussian t-design with N points in R^d yields an unweighted spherical t-design with at most c_tN points.

Background

The paper proves transfers between Gaussian and spherical designs with controlled size when weights are allowed. In particular, an unweighted spherical design can be transferred to an unweighted Gaussian design, but the reverse construction generally produces weights.

A positive answer would allow the projection argument developed for Gaussian designs to preserve unweightedness and would therefore extend the paper's dimensional-projection results to unweighted spherical designs.