General-adversary secret-sharing share-size conjecture

Prove or refute the conjecture that general adversary structures can force some parties in a secret-sharing scheme to receive shares whose size is $2^{\Theta(n)}$ times larger than the secret.

Background

The paper discusses secret sharing as a related secrecy-based task. In contrast to the consensus problems studied in the paper, secret sharing requires parties to obtain shares from which a secret can be reconstructed by sufficiently large quorums.

The cited conjecture asserts an exponential blow-up in share size for some general adversary structures. The paper notes that the conjecture has been proved for certain classes of secret-sharing schemes, but presents it as unresolved in general.