Let X be  a  Hausdorff space such that every real  vector  bundle on X is  summand of  a trivial bundle. Does  this imply that X is  homotopy  equivalent to a  compact  Hausdorf space? This  question is  a "compact  version" of the  following question;

https://mathoverflow.net/questions/149138/paracompactness-and-inner-product-on-vector-bundles