No. If Litecoin completely went away tomorrow, it would have no impact on Logware and Lynx.

Lynx is a code fork of the Litecoin project. The Lynx developers have modified the code fork with Lynx specific enhancements, moving away from the original project objectives of Litecoin. That means whenever Litecoin releases an update, the Lynx development team examines it and only implements specific aspects of it that will benefit Lynx. This may include bug fixes, security enhancements and the occasional feature addition.

This is the nature of open source software. Otherwise, Lynx and Litecoin have no other relationship. Lynx has its own Genesis block, blockchain, and it runs completely separate from Litecoin.