Not quite. Every modern BLAS is (likely) based on Kazushige Goto's implementation, and he was indeed at TACC for a while. But probably the best open source implementation "BLIS" is from UT Austin, but not connected to TACC.
Sort of like BLAS, LAPACK is more than just one implementation. Dongarra described what everybody should do from Tennesse, but other places implemented it elsewhere.