Asymptotic redundancy for binary two-error duplication-correcting codes

Determine the asymptotic behavior of the minimum redundancy ratio r_k^{\min}(n)/\log_2 n for binary codes correcting at most two unrestricted sequential palindromic duplications of a fixed even length k\geq2; in particular, determine whether r_k^{\min}(n)=2\log_2 n+o(\log n), matching the converse bound.

Background

For every fixed even duplication length k\geq2, the paper establishes a binary converse of r_k^{\min}(n)\geq2\log_2 n-(2k+3)-o(1). It also gives a nonconstructive greedy-coloring construction with redundancy at most 4\log_2 n+O_k(1), so the leading coefficient of the optimal redundancy is known only to lie between 2 and 4.

The unresolved issue is whether the converse coefficient 2 is achievable asymptotically. Because even-length reverse-complement duplication is bijectively conjugate to palindromic duplication, an answer for the palindromic channel would transfer to the corresponding reverse-complement channel.