The nonexistence conjecture for perfect binary sequences with parameter $d=1$

For a binary sequence $X$ of length $n$, write $PComS(n,q,c)$ for the family used in the paper, and let a perfect binary sequence have two-level nontrivial autocorrelation with value $d$. In the case $d=1$, integrality gives $n=2u^2+2u+1$ and the relevant family is $PComS(2u^2+2u+1,1,1)$. The nonexistence conjecture for perfect binary sequences with parameter $d=1$. There are no families $PComS(2u^2+2u+1,1,1)$ with $u\geq3$. The paper identifies the cases $u=1$ and $u=2$, corresponding to lengths $5$ and $13$, as the only known perfect binary sequences of this type; the asserted nonexistence for all $u\geq3$ remains open.