  In 1999 Lyngs{\o} and Pedersen proposed a conjecture stating that every
binary circular word of length $n$ with equal number of zeros and ones has an
antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No
progress over a trivial $\frac{1}{2}n$ bound has been achieved since then. We
suggest a palindromic counterpart to this conjecture and provide a non-trivial
infinite series of circular words which prove the upper bound of $\frac{2}{3}n$
for both conjectures at the same time. The construction also works for words
over an alphabet of size $k$ and gives rise to a generalization of the
conjecture by Lyngs{\o} and Pedersen. Moreover, we discuss some possible
strengthenings and weakenings of the named conjectures. We also propose two
similar conjectures for linear words and provide some evidences for them.
