Littlewood-admissibility conjecture for pairs with two interior zeros

Let $n\geq 1$, and let $(k,n)$ be a valid pair for Littlewood polynomials. A pair is Littlewood-admissible if some Littlewood polynomial of degree $n$ has exactly $k$ zeros in the open unit disk and no zeros on the unit circle. Littlewood-admissibility conjecture. The pairs $(2,n)$ and $(n-2,n)$ with $n\equiv 1\pmod 6$ and $n\geq 13$ are the only non-trivial Littlewood-inadmissible pairs. Computations establish this classification for $n\leq31$, while the assertion for all larger degrees remains open.