Is there a single-use signature scheme, where a second use of the private key discloses it to the world?
Trivial solution: generate a random $k$ as part of the private key and include $r$ as part of the public key.

The verifier uses $r$ from public key, so the signer must use the same $k$ for every valid signature. The signer could create multiple related public keys and reuse $D_A$, but then, they might as well just create multiple key-pairs in the first place.