We consider one of the simplest and best known buffer management architectures: the shared memory switch with multiple output queues and uniform packets. It was one of the first models studied by competitive analysis, with the Longest Queue Drop (LQD) buffer management policy shown to be at least $\sqrt{2}$- and at most $2$-competitive; a general lower bound of $4/3$ has been proven for all deterministic online algorithms. Closing the gap between $\sqrt{2}$ and $2$ has remained an open problem in competitive analysis for more than a decade, with only marginal success in reducing the upper bound of $2$. In this work, we first present a simplified proof for the $\sqrt{2}$ lower bound for LQD and then, using a reduction to the continuous case, improve the general lower bound for all deterministic online algorithms from $\frac 43$ to $\sqrt{2}$. Then, we proceed to improve the lower bound of $\sqrt{2}$ specifically for LQD, showing that LQD is at least $1.44546086$-competitive. We are able to prove the bound by presenting an explicit construction of the optimal clairvoyant algorithm which then allows for two different ways to prove lower bounds: by direct computer simulations and by proving lower bounds via linear programming. The linear programming approach yields a lower bound for LQD of $1.4427902$ (still larger than $\sqrt{2}$).