For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, uniqueness of $L^infty$ solutions is an important open question. One possible method to show emph{ill-posedness} would be to construct solutions via convex integration and so-called $T_N$ configurations. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur, one of the strongest available emph{well-posedness} theories. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, we discuss recent work showing the impossibility, for a large class of $2times2$ systems, of doing convex integration via the use of $T_N$ configurations, for the case $N=4$. Our work applies to every well-known $2times2$ hyperbolic system of conservation laws which verifies the "structural Liu entropy condition," including the $p$-system, isentropic Euler, the equations for an ideal gas or the system of shallow water waves, and two coupled copies of Burgers. This is joint work with László Székelyhidi.