A prosthetic remodeling annuloplasty ring for use in tricuspid or mitral valve repairs to provide support after annuloplasty surgery. The annuloplasty ring includes a relatively rigid core extending around an axis that is discontinuous to define two free ends. A suture-permeable interface surrounding the core includes floppy regions adjacent both free ends of the core. Sutures are used to attach the annuloplasty ring to the annulus, including at least one suture through each of the floppy regions to secure the free ends of the ring and minimize the risk of ring dehiscence, or pull through of the sutures through the annulus tissue. The floppy regions may project from each free end into the gap toward each other, be radially enlarged such as paddle-like extensions, or may comprise outwardly lateral extensions at the free ends of the core. The interface may include an elastomeric sleeve and a fabric covering, wherein the floppy regions may be formed by molded portions of the sleeve with the fabric covering, or just the fabric covering.