Memory-length conjecture for tight IQC convergence rates

Let a first-order algorithm have $T$ steps of memory. For each integer $n\geq 1$, an off-by-$n$ pointwise IQC is the quadratic constraint associated with a pair of iterates $(y_{k-n},y_k)$. Consider the semidefinite program governing the convergence-rate bound obtained from the IQC framework. Memory-length conjecture. For first-order algorithms with $T$ steps of memory, off-by-$n$ pointwise IQCs with $n\leq T$ suffice to obtain the tightest convergence rate in this framework; adding off-by-$n$ pointwise IQCs with $n>T$ to the semidefinite program does not improve the bound. The conjecture is motivated by numerical simulations, while the preceding discussion notes that all off-by-$n$ IQCs need not exactly characterize the underlying nonlinearity. It remains open whether the stated cutoff at the memory length holds for the algorithms covered by the framework.