Lin's conjecture on ternary ideal two-level autocorrelation sequences

Let $n=2m+1$, and let $\alpha$ be a primitive element in $\mathbb{F}_{3^n}$. Let $Tr$ denote the trace map from $\mathbb{F}_{3^n}$ to $\mathbb{F}_3$, and define the ternary sequence $S=\{s_i\}$ by

$$s_i=Tr(\alpha^i+\alpha^{(2\cdot 3^m+1)i}),\qquad i=0,1,2,\ldots.$$

Lin's conjecture. The sequence $S$ has ideal two-level autocorrelation; that is, if its period is $N$, then

$$C_S(\tau)=\sum_{i=0}^{N-1}\omega_3^{s_{i+\tau}-s_i}=-1$$

for every $0<\tau<N$.

The paper claims to prove this conjecture using the second-order multiplexing decimation-Hadamard transform, Stickelberger's theorem, the Teichmüller character, and elementary enumeration of ternary numbers. Thus the conjecture is presented as solved in the source paper.