Short answers, pulled from the story.
Joan Daemen and Paris Kitsos submitted the MOSQUITO algorithm to the eSTREAM project in 2005. Their document published that year outlined specific goals for a dedicated self-synchronizing stream cipher.
Antoine Joux and Frédéric Muller demonstrated fatal flaws in the cipher during 2006. Their conference paper stated that all dedicated Self-Synchronizing Stream Ciphers of the KNOT-MOSQUITO family were subject to differential chosen ciphertext attacks.
Käsper et al successfully broke the MOUSTIQUE variant after it reached the final phases. This event left secure design for efficient self-synchronizing stream ciphers as an open research problem with no other algorithms surviving the scrutiny required by the eSTREAM project standards.
The specification details eight registers of varying lengths within the MOSQUITO cipher structure. One register called CCSR holds exactly 128 bits while others range down to just 3 bits.