Starts a HuggingFace sync of the benchmark and returns its current sync status.

The export configuration (sampleMediaMode, segmentExports) is persisted on the benchmark; a field left null keeps the stored value, and an empty segmentExports list resets to the legacy shape. Every sync rebuilds the whole dataset; an existing README is kept. An asset segment carried into the dataset holds exactly one file per prompt: matchups whose prompt carries no value, or a multi-asset, under a configured asset segment are left out of the export (several files belong in several segments). Responds 400 when a segment export names a key the benchmark's prompt structure does not define, when a column does not match ^[a-z][a-z0-9_]{0,63}$, is used twice or collides with a reserved dataset column (model1, model2, image1, image2, video1, video2, file1, file2, or any name starting with weighted_results_ or detailed_results_), when a key is exported twice, or when a mode contradicts the segment's kind (Text only for text segments, Embed / Url only for asset segments). Responds 409 Conflict when a sync is already in progress for the benchmark, or when a repo name is supplied that differs from the one an earlier successful sync established. The repo name is only honored while the benchmark has never completed a sync.