Do publishers allow green OA to supplementary material?

Question

Most journals allow authors to publish a copy of their own publications on their personal or institutional website ("green Open Access"). Sherpa Romeo summarizes this as "author can archive post-print (ie final draft post-refereeing)"

Does his also cover supplementary material, i.e. research data that has been submitted by the author(s) to the publisher and that is available on the publisher's website, but only after paying for access to the paper.