  In recent years, there appeared a considerable interest in the inverse
spectral theory for functional-differential operators with constant delay. In
particular, it is well known that specification of the spectra of two operators
$\ell_j,$ $j=0,1,$ generated by one and the same functional-differential
expression $-y''(x)+q(x)y(x-a)$ under the boundary conditions
$y(0)=y^{(j)}(\pi)=0$ uniquely determines the complex-valued square-integrable
potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[\pi/2,\pi).$ For many
years, it has been a challenging {\it open question} whether this uniqueness
result would remain true also when $a\in(0,\pi/2).$ Recently, a positive answer
was obtained for the case $a\in[2\pi/5,\pi/2).$ In this paper, we give,
however, a {\it negative} answer to this question for $a\in[\pi/3,2\pi/5)$ by
constructing an infinite family of iso-bispectral potentials. Some discussion
on a possibility of constructing a similar counterexample for other types of
boundary conditions is provided, and new open questions are outlined.
